Algebra I - Center For Teaching &...

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Algebra I

Quadratics

2015-11-04

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• Characteristics of Quadratic Equations

• Graphing Quadratic Equations

• Transforming Quadratic Equations

• Solve Quadratic Equations by Graphing

• Solve Quadratic Equations by Factoring

• Solve Quadratic Equations Using Square Roots

• Solve Quadratic Equations by Completing the Square

• Solve Quadratic Equations by Using the Quadratic Formula

• Key Terms

• Solving Application Problems

• The Discriminant

Table of ContentsClick on the topic to go to that section

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Key Terms

Return to Tableof Contents

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Axis of symmetry: The vertical line thatdivides a parabola intotwo symmetrical halves

Axis of Symmetry

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Maximum: The y-value of thevertex if a < 0 and the parabola opens downward

Minimum: The y-value of thevertex if a > 0 and theparabola opens upward

Parabola: The curve resultof graphing aquadratic equation

(+ a) Max

Min

(- a)

Parabolas

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Quadratic Equation: An equation that can be written in the standard form ax 2 + bx + c = 0. Where a, b and c are real numbers and a does not = 0.

Vertex: The highest or lowest point ona parabola.

Zero of a Function: An x value that makes the function equal zero.

Quadratics

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Characteristics of Quadratic

Equations


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A quadratic equation is an equation of the form ax2 + bx + c = 0 , where a is not equal to 0.

Quadratics

The form ax2 + bx + c = 0 is called the standard form of the quadratic equation.

The standard form is not unique. For example, x2 - x + 1 = 0 can be written as the equivalent equation -x2 + x - 1 = 0.

Also, 4x2 - 2x + 2 = 0 can be written as the equivalent equation 2x2 - x + 1 = 0. Why is this equivalent?

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Practice writing quadratic equations in standard form: (Simplify if possible.)

Write 2x2 = x + 4 in standard form:

Writing Quadratic Equations

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1 Write 3x = -x2 + 7 in standard form:

A. x2 + 3x-7= 0

B. x2 -3x +7=0

C. -x2 -3x -7= 0 Ans

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2 Write 6x2 - 6x = 12 in standard form:

A. 6x2 - 6x -12 = 0

B. x2 - x - 2 = 0

C. -x2 + x + 2 = 0

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3 Write 3x - 2 = 5x in standard form:

A. 2x + 2 = 0

B. -2x - 2 = 0

C. not a quadratic equation

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← upward→downward

The graph of a quadratic is a parabola, a u-shaped figure.

The parabola will open upward or downward.

Characteristics of Quadratic Functions

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A parabola that opens upward contains a vertex that is a minimum point. A parabola that opens downward contains a vertex that is a maximum point.

vertex

vertex


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The domain of a quadratic function is all real numbers.


D = Reals

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To determine the range of a quadratic function,ask yourself two questions:

> Is the vertex a minimum or maximum? > What is the y-value of the vertex?

If the vertex is a minimum, then the range is all real numbers greater than or equal to the y-value.

The range of this quadratic is [–6,∞ )


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If the vertex is a maximum, then the range is all real numbers less than or equal to the y-value.

The range of this quadratic is (–∞ ,10]


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7. An axis of symmetry (also known as a line of symmetry) will divide the parabola into mirror images. The line of symmetry is always a vertical line of the form

x=2


x = –b2a

x = –(–8) 2(2)

= 2 y = 2x2 – 8x + 2

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To find the axis of symmetry simply plug the values of a and b into the equation: Remember the form ax 2 + bx + c. In this example a = 2, b = -8 and c =2


x=2x = –b2a

x = –(–8) 2(2)

= 2 y = 2x2 – 8x + 2

a b c

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The x-intercepts are the points at which a parabola intersects the x-axis. These points are also knownas zeroes, roots or solutions and solution sets. Each quadratic equation will have two, one or no real x-intercepts.


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4 The vertical line that divides a parabola into two symmetrical halves is called...

A discriminant

B perfect square

C axis of symmetry

D vertex

E slice

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5 What are the vertex and axis of symmetry of the parabola shown in the diagram below?A vertex: (1,−4);

axis of symmetry: x = 1B vertex: (1,−4);

axis of symmetry: x = −4

C vertex: (−4,1); axis of symmetry: x = 1

D vertex: (−4,1); axis of symmetry: x = −4

From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011.

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6 The equation y = x 2 + 3x − 18 isgraphed on the set of axes below.

−3 and 60 and −183 and −6 3 and −18

A B C D

Based on this graph, what are the roots of the equation x2 + 3x − 18 = 0?

From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from

www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011.

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C

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7 The equation y = − x 2 − 2x + 8 isgraphed on the set of axes below.

Based on this graph, what are the roots of the equation −x2 − 2x + 8 = 0?

A 8 and 0

B 2 and –4

C 9 and –1

D 4 and –2



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8 What is an equation of the axis of symmetry of the parabola represented by y = −x2 + 6x − 4?

A x = 3B y = 3C x = 6D y = 6



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9 The height, y, of a ball tossed into the air can be represented by the equation y = –x2 + 10x + 3, where x is the elapsed time. What is the equation of the axis of symmetry of this parabola?

A y = 5

B y = –5

C x = 5

D x = –5



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5 6 7 8 9 10

18 16 14 12 10

8642



10 What is the equation of the axis of symmetry of the parabola shown in the diagram below?

A x = –0.5

B x = 2

C x = 4.5D x = 13 A

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B

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TransformingQuadratic Equations


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x x2

-3 9-2 4-1 10 01 12 43 9

The quadratic parent equation is y = x2. The graph of all other quadratic equations are transformations of the graph of y= x2.

Quadratic Parent Equation

y = x2

y = – – x223

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The quadratic parent equation is y = x 2. How is y = x2 changed into y = 2x2?

x 2

-3 18

-2 8

-1 2

0 0

1 2

2 8

3 18


y = 2x2

y = x2

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x 0.5

-3 4.5

-2 2

-1 0.5

0 0

1 0.5

2 2

3 4.5

The quadratic parent equation is y = x 2. How is y = x2 changed into y = .5x2?


y = x2

y = – x212

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How does a > 0 affect the parabola? How does a < 0 affect the parabola?

What does "a" do in y = ax2+ bx + c ?What Does "A" Do?

y = x2 y = –x2

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What does "a" also do in y =ax2 + bx +c ?

How does your conclusion about "a" changeas "a" changes?

What Does "A" Do?

y = x2y = – x212

y = 3x2

y = –1x2 y = –3x2 y = – – x212

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If the absolute value of a is > 1, then the graph of the equation is narrower than the graph of the parent equation.

If the absolute value of a is < 1, then the graph of the equation is wider than the graph of the parent equation.

If a > 0, the graph opens up.

If a < 0, the graph opens down.

What does "a" do in y = ax2 + bx + c ?

What Does "A" Do?

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11 Without graphing determine which direction does the parabola open and if the graph is wider or narrower than the parent equation.

A up, wider

B up, narrower

C down, wider

D down, narrower

Ans

wery = .3x2

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12 Without graphing determine which direction does the parabola open and if the graph is wider or narrower than the parent function.

A up, wider

B up, narrower

C down, wider

D down, narrower

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y = –4x2

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13 Without graphing determine which direction does the parabola open and if the graph is wider or narrower than the parent equation.

A up, wider

B up, narrower

C down, wider

D down, narrower

y = –2x2 + 100x + 45

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A up, wider

B up, narrower

C down, wider

D down, narrower

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y = – – x223

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A up, wider

B up, narrower

C down, wider

D down, narrower

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B

y = – – x275

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What does "c" do in y = ax2 + bx + c ?

What Does "C" Do?

y = x2 + 6

y = x2 + 3

y = x2

y = x2 – 2

y = x2 – 5

y = x2 – 9

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"c" moves the graph up or down the same value as "c."

"c" is the y- intercept.

What does "c" do in y = ax2 + bx + c ?

What Does "C" Do?

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16 Without graphing, what is the y- intercept of the the given parabola?

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y = x2 + 17

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y = –x2 –6

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y = –3x2 + 13x – 9

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y = 2x2 + 5x

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20 Choose all that apply to the following quadratic:

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opens up

opens down

wider than parent function

narrower than parent

function

A

B

C

D

y-intercept of y = –4

y-intercept of y = –2

y-intercept of y = 0




A

B

C

D

E

F

f(x) = –.7x2 –4

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21 Choose all that apply to the following quadratic:

A opens up

B opens down

C wider than parent function

D narrower than parent function

E y-intercept of y = –4

F y-intercept of y = –2

G y-intercept of y = 0

H y-intercept of y = 2I y-intercept of y = 4

J y-intercept of y = 6

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f(x) = – – x2 –6x43

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22 The diagram below shows the graph of y = −x 2 − c.

A B C D

Which diagram shows the graph of y = x2 − c?



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Graphing Quadratic Equations


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Graph by Following Six Steps:

Step 1 - Find Axis of Symmetry

Step 2 - Find Vertex

Step 3 - Find Y intercept

Step 4 - Find two more points

Step 5 - Partially graph

Step 6 - Reflect

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Axis of Symmetry

Axis of SymmetryStep 1 - Find Axis of Symmetry

What is the Axis of Symmetry?

Teac

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otes The line that runs down

the center of a parabola. This line divides the

graph into two perfect halves.

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Graph y = 3x2 – 6x + 1

Formula:a = 3b = –6x = – (–6) = 6 = 1 2(3) 6

x = –––b2a

The axis of symmetryis x = 1.

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y = 3(1)2 + –6(1) + 1y = 3 – 6 + 1y = –2Vertex = (1,–2)


Step 2 - Find the vertex by substituting the value of x(the axis of symmetry) into the equation to get y.

y = 3x2 – 6x + 1

a = 3, b = –6 and c = 1

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Step 3 - Find y interceptWhat is the y-intercept?

y- intercept

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The point where the line passes through they-axis. This occurs

when the x-value is 0.

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The y- intercept is always the c value, because x = 0.

c = 1

The y-intercept is 1 and the graph passes through (0,1).

y = ax2 + bx + c

y = 3x2 – 6x + 1

Step 3 - Find y intercept

Graph y = 3x2 – 6x + 1

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Choose different values of x and plug in to find points.

Step 4 - Find Two More Points

Find two more points on the parabola.

Graph y = 3x2 – 6x + 1

Let's pick x = –1 and x = –2

y = 3x2 – 6x + 1

y = 3(–1)2 – 6(–1) + 1

y = 3 + 6 + 1y = 10

(–1,10)

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Step 4 - Find Two More Points (continued)Graph y = 3x2 – 6x + 1

y = 3x2 – 6x + 1

y = 3(–2)2 – 6(–2) + 1

y = 3(4) + 12 + 1

y = 25

(–2, 25)

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Step 5 - Graph the Axis of Symmetry

Graph the axis of symmetry, the vertex, the point containing the y-intercept and two other points.

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(4,25)

Step 6 - Reflect the Points

Reflect the points across the axis of symmetry. Connect the points with a smooth curve.

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23 What is the axis of symmetry for y = x2 + 2x - 3 (Step 1)?

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24 What is the vertex for y = x2 + 2x - 3 (Step 2)?

A (-1, -4)

B (1, -4)

C (-1, 4)

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25 What is the y-intercept for y = x2 + 2x - 3 (Step 3)?

A -3

B 3

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axis of symmetry = –1

vertex = –1, –4

y intercept = –3

2 other points (step 4)

(1,0)

(2,5)

Partially graph (step 5)

Reflect (step 6)

Graphy= x2 + 2x – 3

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Graphy = 2x2 – 6x + 4

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Graph

y = –x2 – 4x + 5

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Graphy = 3x2 – 7

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Solve Quadratic Equations by Graphing


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Find the Zeros

One way to solve a quadratic equation in standard form is find the zeros by graphing.

A zero is the point at which the parabola intersects the x-axis.

A quadratic may have one, two or no zeros.

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No zeroes(doesn't crossthe "x" axis)

2 zeroes; x = -1 and x=3

1 zero;x=1

Find the ZerosHow many zeros do the parabolas have?

What are the values of the zeros?

click click click

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Review



Step 3 - Find Y intercept

Step 4 - Find two more points

Step 5 - Partially graph

Step 6 - Reflect

To solve a quadratic equation by graphing follow the6 step process we already learned.

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Which of these is in standard form?

y = 2x2 – 12x + 18

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Solve the equation by graphing.

–12x + 18 = –2x2

y = –2x2 – 12x + 18

y = –2x2 + 12x – 18

B

A

C

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What is the axis of symmetry?

y = –2x2 + 12x – 18

A –3B 3C 4D –5

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y = –2x2 + 12x – 18

What is the vertex?

A (3,0)

B (–3,0)

C (4,0)

D (–5,0)

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What is the y- intercept?

A (0, 0)

B (0, 18)

C (0, –18)

D (0, 12)

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C

y = –2x2 + 12x – 18

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A B

C D

If two other points are (5, –8) and (4 ,–2),what doesthe graph of y = –2x2 + 12x – 18 look like?

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31 y = –2x2 + 12x – 18

What is(are) the zero(s)?

A –18

B 4

C 3

D –8

click for graph of answer

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Solve Quadratic Equations by Factoring


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Review of factoring - To factor a quadratic trinomial of the form x2 + bx + c, find two factors of c whose sum is b.

Example - To factor x 2 + 9x + 18, look for factors whose sum is 9.

Factors of 18 Sum

1 and 18 19

2 and 9 11

3 and 6 9

Solving Quadratic Equationsby Factoring

x2 + 9x + 18 = (x + 3)(x + 6)

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When c is positive, it's factors have the same sign.The sign of b tells you whether the factors are positive or negative.When b is positive, the factors are positive.When b is negative, the factors are negative.


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4. Multiply the Last terms (x + 3)(x + 2) 3 2 = 6

3. Multiply the Inner terms (x + 3)(x + 2) 3 x = 3x

2. Multiply the Outer terms (x + 3)(x + 2) x 2 = 2x

1. Multiply the First terms (x + 3)(x + 2) x x = x2

F O I L

Remember the FOIL method for multiplying binomials


(x + 3)(x + 2) = x2 + 2x + 3x + 6 = x2 + 5x + 6

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For all real numbers a and b, if the product of two quantities equals zero, at least one of the quantities equals zero.

Numbers Algebra

3(0) = 0 If ab = 0,

4(0) = 0 Then a = 0 or b = 0

Zero Product Property

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Example 1: Solve x2 + 4x – 12 = 0

x + 6 = 0 or x – 2 = 0 –6 –6 + 2 +2 x = –6 x = 2

–62 + 4(–6) – 12 = 0 –62 + (–24) – 12 = 0 36 – 24 – 12 = 0 0 = 0 or22 + 4(2) – 12 = 0 4 + 8 – 12 = 0 0 = 0

Use "FUSE" !


Factor the trinomial using the FOIL method.

Use the Zero property

Substitue found value into original equation

Equal - problem solved! The solutions are -6 and 2.

(x + 6) (x – 2) = 0

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Example 2: Solve x2 + 36 = 12x –12x –12x

The equation has to be written in standard form (ax2 + bx + c). So subtract 12x from both sides.


Factor the trinomial using the FOIL method.

Use the Zero property


Equal - problem solved!

x2 – 12x + 36 = 0

(x – 6)(x – 6) = 0 x – 6 = 0 +6 +6 x = 6

62 + 36 = 12(6)

36 + 36 = 72

72 = 72

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Example 3: Solve x2 – 16x + 48= 0

(x – 4)(x – 12) = 0

x – 4 = 0 x –12 = 0 +4 +4 +12 +12 x = 4 x = 12


Factor the trinomial using the FOIL method. Use the Zero property


Equal - problem solved!

42 – 16(4) + 48 = 0 16 – 64 + 48 = 0 –48+48 = 0 0 = 0

122 – 16(12) + 48 = 0 144 –192 + 48 = 0 –48 + 48 = 0 0 = 0

–48

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32 Solve

A –7

B –5

C –3

D –2

E 2

F 3

G 5

H 6

I 7

J 15

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x2 – 5x + 6 = 0

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33 Solve m2 + 10m + 25 = 0

A –7

B –5

C –3

D –2

E 2

F 3

G 5

H 6

I 7

J 15

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34 Solve h2 – h = 12

A –12

B –4

C –3

D –2

E 2

F 3

G 4

H 6

I 8J 12

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35 Solve d2 – 35d = 2d

A –7

B –5

C –3

D 35E 12

F 0

G 5

H 6

I 7J 37

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36 Solve 8y2 + 2y = 3

A –3/4

B –1/2

C –4/3D –2

E 2

F 3/4

G 1/2

H 4/3

I –3

J 3

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37 Which equation has roots of −3 and 5?

A x2 + 2x − 15 = 0B x2 − 2x − 15 = 0C x2 + 2x + 15 = 0 D x2 − 2x + 15 = 0 A

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Solve Quadratic Equations Using Square Roots


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You can solve a quadratic equation by the square root method if you can write it in the form:

x² = c

If x and c are algebraic expressions, then:

x = c or x = – c

written as:

x = ± c

Square Root Method

√ √

√

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Solve for z: z² = 49z = ± 49z = ±7

The solution set is 7 and –7

Square Root Method

√

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The solution set is and –

A quadratic equation of the form x2 = c can be solved using the Square Root Property.

Example: Solve 4x2 = 20

x = ±

Square Root Method

4x2 = 20 4 4 x2 = 5

Divide both sides by 4 to isolate x²

5√ 5√

5√

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5x2 = 20 5 5

x2 = 4

x = or x = – x = ± 2

4 4

Square Root Method

Solve 5x² = 20 using the square root method:

√ √

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2x – 1 = 202x – 1 = (4)(5)2x – 1 = 2 52x = 1 + 2 5 1 + 2 5x = 2

Solve (2x – 1)² = 20 using the square root method. or

Square Root Method

2x – 1 = – 202x – 1 = – (4)(5)2x – 1 = –2 52x = 1 – 2 5 1 – 2 5x = 2

solution: x = 1 ± 2 5

2

√

√√√√

√

√√√√

√click

click click

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38 When you take the square root of a real number, your answer will always be positive.

True

False

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False

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39 If x2 = 16, then x =

A 4

B 2

C –2

D 26

E –4

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40 If y2 = 4, then y =

A 4

B 2

C –2

D 26

E –4

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B, C

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41 If 8j2 = 96, then j =

Ans

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A – 3 2

B – 2 3

C 2 3

D 3 2

E ±12

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42 If 4h 2 –10= 30, then h =

Ans

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A – 10

B – 2 5

C 2 5

D 10

E ±10

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43 If (3g – 9) 2 + 7= 43, then g =

Ans

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A, D

A 1

B 9 – 5 2

C 9 + 5 2

D 5

E ±3

3

3

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Solving Quadratic Equations by

Completing the Square


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x2 + 8x + ___

x2 + 20x + 100

x2 – 16x + 64

x2 – 2x + 1

Before we can solve the quadratic equation, we first have to find the missing value of C. To do this, simply take the value of b, divide it in 2 and then square the result.

Find the value that completes the square.

(b/2)2

8/2 = 4

42 = 16

ax2+bx+c

Find the Missing Value of "C"

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44 Find (b/2)2 if b = 14

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107

45 Find (b/2)2 if b = –12

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46 Complete the square to form a perfect square t rinomial

x2 + 18x + ?

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47 Complete the square to form a perfect square trinomial

x2 – 6x + ?

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Solving Quadratic Equationsby Completing the Square

Step 1 - Write the equation in the form x 2 + bx = c

Step 2 - Find (b ÷ 2)2

Step 3 - Complete the square by adding (b ÷ 2)2 to both sides of the equation.

Step 4 - Factor the perfect square trinomial.

Step 5 - Take the square root of both sides

Step 6 - Write two equations, using both the positive and negative square root and solve each equation.

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Let's look at an example to solve: x2 + 14x = 15

x2 + 14x = 15 Step 1 - Already done!

(14 ÷ 2)2 = 49 Step 2 - Find (b÷2) 2

x2 + 14x + 49 = 15 + 49 Step 3 - Add 49 to both sides

(x + 7)2 = 64 Step 4 - Factor and simplify

x + 7 = ±8 Step 5 - Take the square root of both sides

x + 7 = 8 or x + 7 = –8 Step 6 - Write and solve two equations

x = 1 or x = –15


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Another example to solve: x 2 – 2x – 2 = 0

x2 – 2x – 2 = 0 Step 1 - Write as x 2+bx=c +2 +2x2 – 2x = 2

(–2 ÷ 2)2 = (–1)2 = 1 Step 2 - Find (b÷2)2

x2 – 2x + 1 = 2 + 1 Step 3 - Add 1 to both sides

(x – 1)2 = 3 Step 4 - Factor and simplify

x – 1 = ± 3 Step 5 - Take the square root of both sides

x – 1 = 3 or x – 1 = – 3 Step 6 - Write and solve two equationsx = 1 + 3 or x = 1 – 3


√

√ √

√ √

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48 Solve the following by completing the square :

x2 + 6x = –5

A –5

B –2C –1D 5E 2

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49 Solve the following by completing the square:

x2 – 8x = 20

A –10B –2C –1D 10E 2

Ans

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115

50 Solve the following by completing the square :

–36x = 3x 2 + 108

A –6

B 6C 0D 6E – 6

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A

116

x – = ±5 43 3

10x 3

x2 – = –1

Write as x 2+bx=c

Find (b÷2)2

Add 25/9 to both sidesFactor and simplify

A more difficult example:Solve3x2 – 10x = –3

3x2 10x = –3 3 3 3

–

10 3

÷ 2 = x = = –10 1 –5 25 3 2 3 9)( ( ) ( )

2 2 2

10x 25 25 3 9 9

x2 – + = –1 +

x – =5 163 9)(

2

√ Take the square root of both sides

Answ

er

x – =

x = 3

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– 254

51 Solve the following by completing the square:

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4x2 – 7x – 2 = 0

– 14

A

B

C

D

E

14

– 254

–

2

118

The Discriminant


119

x = –b ± √b2 – 4ac 2a

Discriminant - the part of the equation under the radical sign in a quadratic equation.

The Discriminant

b2 – 4ac is the discriminant

120

ax2 + bx + c = 0

The discriminant, b2 – 4ac, or the part of the equation under the radical sign, may be used to determine the number of

real solutions there are to a quadratic equation.

The Discriminant

If b2 – 4ac > 0, the equation has two real solutionsIf b2 – 4ac = 0, the equation has one real solutionIf b2 – 4ac < 0, the equation has no real solutions

121

The Discriminant

Remember:The square root of a positive number has two solutions.The square root of zero is 0.

The square root of a negative number has no real solution.

122

Example

√4 = ± 2

(2) (2) = 4 and (–2)(–2) = 4So BOTH 2 and –2 are solutions

The Discriminant

123

What is the relationship between the discriminant of a quadratic and its graph?

Discriminant(8)2 – 4(1)(10) = 64 – 40 = 24 (–6)2 –4(3)(–4) = 36 + 48 = 84

The Discriminant

y = x2 – 8x + 10 y = 3x2 + 8x – 4

124


The Discriminant

Discriminant(–4)2 – 4(2)(2) = 16 – 16 = 0 (6)2 –4(1)(9) = 36 – 36 = 0

y = 2x2 – 4x + 2 y = x2 + 6x + 9

125


Discriminant

(5)2 – 4(1)(9) = 25 – 36 = –11 (–3)2 –4(3)(4) = 9 – 48 = –39

The Discriminant

y = x2 + 5x + 9 y = 3x2 – 3x + 4

126

52 What is value of the discriminant of

2x2 – 3x + 5 = 0 ?

Ans

wer

127

53 Find the number of solutions using the discriminant for 2x2 – 3x + 5 = 0

A 0

B 1

C 2 Ans

wer

128

54 What is value of the discriminant of

x 2 – 8x + 4 = 0 ?

Ans

wer

129

55 Find the number of solutions using the discriminant for x2 – 8x + 4 = 0

A 0

B 1

C 2 Ans

wer

130

Solve Quadratic Equations by Using

the Quadratic Formula


131

At this point you have learned how to solve quadratic equations by:· graphing· factoring· using square roots and · completing the square

Today we will be given a tool to solve ANY quadratic equation.

It ALWAYS works.

Many quadratic equations may be solved using these methods; however, some cannot be solved using any of these methods.

Solve Any Quadratic Equation

132

"x equals the opposite of b, plus or minus the square root of b squared minus 4ac, all divided by 2a."

The Quadratic Formula

The solutions of ax2 + bx + c = 0, where a ≠ 0, are:

x = –b ± b2 – 4ac√2a

133

x = –3 ± √32 –4(2)(–5) 2(2)

x = –b ± √b2 –4ac 2a

continued on next slide

Write the Quadratic Formula

Identify values of a, b and c

Substitute the values of a, b and c

2x2 + 3x + (–5) = 0

2x2 + 3x – 5 = 0

Example 1


134

x = –3 – 7 4

= –3 ± 7 4

x = –3 ± √49 4

x = –3 ± √9 – (–40) 4

x = –3 + 7 4 or


Simplify

Write as two equations

Solve each equationx = 1 or x = –5 2

135

Solution on next slide

The Quadratic FormulaExample 2 2x = x2 – 3

Remember - In order to use the Quadratic Formula, the equation must be in standard form (ax2 + bx +c = 0).

First, rewrite the equation in standard form.

2x = x2 – 3–2x –2x

0 = x2 + (-2x) + (–3)

x2 + (–2x) + (–3) = 0

Use only addition for standard form

Flip the equation

Now you are ready to use the Quadratic Formula

136

x = –(–2) ± √(–2)2 –4(1)(–3) 2(1)

x = –b ± √b2 –4ac 2a

Continued on next slide

The Quadratic Formulax2 + (–2x) + (–3) = 0

1x2 + (–2x) + (–3) = 0 Identify values of a, b and c



137

x = 2 ± √162

= 2 ± 42

x = 2 ± √4 – (–12)2a

Simplify

x = 2 ± 42

or x = 2 - 42

x = 3 or x = –1

Write as two equations

Solve each equation


138

56 Solve the following equation using the quadratic formula:

A -5

B -4

C -3

D -2

E -1

F 1

G 2

H 3

I 4

J 5

Ans

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x2 – 5x + 4 = 0

139


A –5

B –4

C –3

D –2

E –1

F 1

G 2

H 3I 4J 5

Ans

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x2 = x + 20

140

–3 2


A –5

B –4

C

D –2

E –1

F 1

G 2

H

I 4

J 5

Ans

wer

2x2 + 12 = 11x

32

141

x = -b ± √b2 -4ac

2a


The Quadratic FormulaExample 3

x2 – 2x – 4 = 0

1x2 + (–2x) + (–4) = 0 Identify values of a, b and c



x = –(–2) ± √(–2)2 –4(1)(–4) 2(1)

142


x = 2 ± √202

x = 2 ± √4 – (–16)2

Simplify

x = 2 ± 2√52

Write as two equationsor x = 2 - 2√52

orx = 2 ± √202

x = 2 - √202

x = 1 + √5 or x = 1 – √5

x ≈ 3.24 or x ≈ –1.24 Use a calculator to estimate x

143

59 Find the larger solution to

Ans

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x2 + 6x – 1 = 0

144

60 Find the smaller solution to

Ans

wer

x2 + 6x – 1 = 0

145

Application Problems


146

A sampling of applied problems that lend themselves to being solved by quadratic equations:

Number Reasoning

Free Falling Objects

DistancesGeometry: Dimensions

Height of a Projectile

Quadratic Equations and Applications

147

The product of two consecutive negative integers is 1,122. What are the numbers?

Remember that consecutive integers are one unit apart, so the numbers are n and n + 1.

Multiplying to get the product:n(n + 1) = 1122 n2 + n = 1122 n2 + n – 1122 = 0 (n + 34)(n - 33) = 0

n = –34 and n = 33.

The solution is either –34 and –33 or 33 and 34, since the directionask for negative integers –34 and –33 are the correct pair.

→ STANDARD Form→ FACTOR

Number Reasoning

148

PLEASE KEEP THIS IN MIND

When solving applied problems that lead to quadratic equations, you might get a solution that does not satisfy the physical constraints of the problem.

For example, if x represents a width and the two solutions of the quadratic equations are –9 and 1, the value –9 is rejected since a width must be a positive number.


149

61 The product of two consecutive even integers is 48. Find the smaller of the two integers.

Hint: x(x+2) = 48

Ans

werClick to reveal hint

150

The product of two consecutive integers is 272.

What are the numbers?

TRY THIS:


151

The product of two consecutive even integers is 528. What is the smaller number?

62

Ans

wer

152

The product of two consecutive odd integers is 1 less than four times their sum. Find the two integers.

Let n = 1st number n + 2 = 2nd number

More of a challenge...

n(n + 2) = 4[n + (n + 2)] – 1n2 + 2n = 4[2n + 2] – 1n2 + 2n = 8n + 8 – 1n2 + 2n = 8n + 7n2 – 6n - 7 = 0(n – 7)(n + 1) = 0

n = 7 and n = –1

Which one do you use?Or do you use both?

153

More of a challenge...

If n = 7 then n + 2 = 9

7 x 9 = 4[7 + (7 + 2)] – 163 = 4(16) – 163 = 64 – 163 = 63

If n = –1 then n + 2 = –1 + 2 = 1

(–1) x 1 = 4[–1 + (–1 + 2)] – 1–1 = 4[–1 + 1] – 1–1 = 4(0) – 1–1 = –1

We get two sets of answers.

154

63 The product of a number and a number 3 more than the original is 418. What is the smallest value the original number can be?

Ans

wer

155

64 Find three consecutive positive even integers such that the product of the second and third integers is twenty more than ten times the first integer. Enter th e value of the smaller even integer.

Ans

wer


156

65 When 36 is subtracted from the square of a number, the result is five times the number. What is the positive solution?

A 9B 6C 3D 4


Ans

wer

157

66 Tamara has two sisters. One of the sisters is 7 years older than Tamara.The other sister is 3 years younger than Tamara. The product of Tamara’s sisters’ ages is 24. How old is Tamara?

Ans

wer


158

Two cars left an intersection at the same time, one heading north and one heading west. Some time later, they were exactly 100 miles apart. The car that headed north had gone 20 miles farther than the car headed west. How far had each car traveled?

Step 1 - Read the problem carefully.

Step 2 - Illustrate or draw your information.x+20

100

x

Example

Step 3 - Assign a variableLet x = the distance traveled by the car heading westThen (x + 20) = the distance traveled by the car heading northStep 4 - Write an equationDoes your drawing remind you of the Pythagorean Theorem? a2 + b2 = c2 Continued on next slide

Distance Problems

159

Step 5 - Solve a2 + b2 = c2

x2 + (x+20)2 = 1002

x2 + x2 + 40x + 400 = 10,000

2x2 + 40x – 9600 = 0

2(x2 +20x – 4800) = 0

x2 + 20x – 4800 = 0

100x+20

x

Square the binomial

Standard form

Factor

Divide each side by 2

Think about your options for solving the rest of this equation. Completing the square? Quadratic Formula?


Distance Problems

160

x = –20 ±√400 – 4(1)(–4800) 2

Distance ProblemsDid you try the quadratic formula?

x = –20 ±√19,600 2

x = 60 or x = -80

Since the distance cannot be negative, discard the negative solution. The distances are 60 miles and 60 + 20 = 80 miles.

Step 6 - Check your answers.

161

67 Two cars left an intersection at the same time,one heading north and the other heading east. Some time later they were 200 miles apart. If the car heading east traveled 40 miles farther, how far did the car traveling north go?

Ans

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162

x

x + 6

Geometry Applications

Area ProblemThe length of a rectangle is 6 inches more than its width. The area of the rectangle is 91 square inches. Find the dimensions of the rectangle.

Step 1 - Draw the picture of the rectangle. Let the width = x and the length = x + 6

Step 2 - Write the equation using theformula Area = length x width

163

Step 3 - Solve the equation

x( x + 6) = 91

x2 + 6x = 91

x2 + 6x – 91 = 0

(x – 7)(x + 13) = 0

x = 7 or x = –13

Since a length cannot be negative...

The width is 7 and the length is 13.

Geometry Applications

164

68 The width of a rectangular swimming pool is 10 feet less than its length. The surface area of the pool is 600 square feet. What is the pool's width?

Hint: (L)(L – 10) = 600. Ans

wer

Click to reveal hint

165

69 A square's length is increased by 4 units and its width is increased by 6 units. The result of this transformation is a rectangle with an area that 195 square units. Find the area of the original square.

Ans

wer

166

length

x

x

70 The rectangular picture frame below is the same width all the way around. The photo it surrounds measures 17" by 11". The area of the frame and photo combined is 315 sq. in. What is the length of the outer frame?

Answ

er

167

71 The area of the rectangular playground enclosure at South School is 500 square meters. The length of the playground is 5 meters longer than the width. Find the dimensions of the playground, in meters.[Only an algebraic solution will be accepted.]

Answ

er

From the New York State Education Department. Office of Assessment Policy,

Development and Administration. Internet. Available fromwww.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011.

168

72 Jack is building a rectangular dog pen that he wishes to enclose. The width of the pen is 2 yards less than the length. If the area of the dog pen is 15 square yards, how many yards of fencing would he need to completely enclose the pen?

Answ

er

169

Free Falling Objects Problems

170

73

Answ

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A person walking across a bridge accidentally drops an orange in the river below from a height of 40 ft. The function h = –16t2 + 40 gives the orange's approximate height h above the water, in feet, after t seconds. In how many t seconds will the orange hit the water? (Round to the nearest tenth.)

Hint: when it hits the water it is at 0.

171

74 Greg is in a car at the top of a roller-coaster ride. The distance, d, of the car from the ground as the car descends is determined by the equation d = 144 – 16t 2 where t is the number of seconds it takes the car to travel down to each point on the ride. How many seconds will it take Greg to reach the ground?

Answ

er

172

75 The height of a golf ball hit into the air is modeled by the equation h = –16t2 + 48t, where h represents the height, in feet, and t represents the number of seconds that have passed since the ball was hit. What is the height of the ball after 2 seconds?

Answ

er

From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17,

June, 2011.

A 16 ft

B 32 ft

C 64 ft

D 80 ft

173

Height of Projectiles Problems

174

76 A skyrocket is shot into the air. It's altitude in feet, h, after t seconds is given by the functionh = –16 t2 + 128 t.

What is the rocket's maximum altitude?

Answ

er

175

77 A rocket is launched from the ground and follows a parabolic path represented by the equation y = –x 2 + 10x. At the same time, a flare is launched from a height of 10 feet and follows a straight path represented by the equation y = –x + 10. Using the accompanying set of axes, graph the equations that represent the paths of the rocket and the flare, and find the coordinates of the point or points where the paths intersect.

From the New York State Education Department. Office of Assessment Policy,

Development and Administration. Internet. Available fromwww.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011.

Answ

er

Algebra I - Center For Teaching &...

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