Aim: Completing the Square Course: Adv. Alg. & Trig Aim: How do we solve quadratic equations by...

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Aim: Completing the Square Course: Adv. Alg. & Trig Aim: How do we solve quadratic equations by completing the square? An archer shoots an arrow into the air with an initial velocity of 128 feet per second. Because speed is the absolute value of velocity, the arrow’s speed, s, in feet per second, after t seconds is | -32t + 128 |. Find the values of t for which s is less that 48 feet per second. s = | -32t + 128 | < 48 Rewrite into 2 derived inequalities x > 2.5 x < 5.5 Solve each inequality Check your answers -32(3) + 128 < 48 -32(5) + 128 > - -32 > -4 True ! 32 < 48 0 1234567 - 7 - 6 - 5 - 4 - 3 - 2 - 1 -32t + 128 < 48 -32t + 128 > -4 or

Transcript of Aim: Completing the Square Course: Adv. Alg. & Trig Aim: How do we solve quadratic equations by...

Page 1: Aim: Completing the Square Course: Adv. Alg. & Trig Aim: How do we solve quadratic equations by completing the square? An archer shoots an arrow into.

Aim: Completing the Square Course: Adv. Alg. & Trig

Aim: How do we solve quadratic equations by completing the square?

An archer shoots an arrow into the air with an initial velocity of 128 feet per second. Because speed is the absolute value of velocity, the arrow’s speed, s, in feet per second, after t seconds is | -32t + 128 |. Find the values of t for which s is less that 48 feet per second.

s = | -32t + 128 | < 48

Rewrite into 2 derived inequalities

x > 2.5 x < 5.5Solve each inequality

Check your answers -32(3) + 128 < 48 -32(5) + 128 > -48

-32 > -48True! 32 < 48

0 1 2 3 4 5 6 7-7 -6 -5 -4 -3 -2 -1

-32t + 128 < 48 -32t + 128 > -48or

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Aim: Completing the Square Course: Adv. Alg. & Trig

John is changing the floor plan of his home to include the dining room. The current dimensions of the room are 13’ by 13’. John wants to keep the square shape of the room and increase to total floor space to 250 square feet. How much will this add to the dimensions of the current room?

Let x = added lengthto each side of room

13’

13’ x

x

(13 + x)2 = 250 A = s2

(13 x)2 250

13 x 250

x 250 13

s

s

Quadratic Equation Problem

2.8

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Aim: Completing the Square Course: Adv. Alg. & Trig

Aim: How do we solve quadratic equations by completing the square?

Evaluate a0 + a1/3 + a -2 when a = 8

Do Now:

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Aim: Completing the Square Course: Adv. Alg. & Trig

Evaluating

Evaluate a0 + a1/3 + a -2 when a = 8

80 + 81/3 + 8-2 replace a with 8

1 + 81/3 + 8-2 x0 = 1

x1/3 =

x3

83 21 + 2 + 8-2

x–n = 1/xn 8–2 = 1/82 = 1/641 + 2 + 1/64

3 1/64 combine like terms

If m = 8, find the value of (8m0)2/3

(8 • 80)2/3 replace m with 8

(8)2/3

(8 • 1)2/3 x0 = 1

= 4

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Aim: Completing the Square Course: Adv. Alg. & Trig

Simplifying – Fractional Exponents

A rational expression that contains a fractional exponent in the denominator must also be rationalized. When you simplify an expression, be sure your answer meets all of the given conditions.

Conditions for a Simplified Expression1. It has no negative exponents.2. It has no fractional exponents in the

denominator.3. It is not a complex fraction.4. The index of any remaining radical is

as small as possible.

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Aim: Completing the Square Course: Adv. Alg. & Trig

Simplifying – Fractional Exponents

3 5

4 4m n

1 1 5

3 6 12a b c

3 42 2

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Aim: Completing the Square Course: Adv. Alg. & Trig

Completing the Square

Square of Binomial Perfect Square Trinomial

(x + 3)2 = x2 + 6x + 9

(x - 4)2 = x2 - 8x + 16

(x - c)2 = x2 - 2cx + c2

(x - 7)2 = x2 - 14x + 49

In a perfect square, there is a relationshipbetween the coefficient of the middle termand the constant (3rd) term. Describe it.

Find the value of the c that makes x2 + 18x + c a perfect square.

x2 + 18x + 81 = (x + 9)2

c = 81

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Aim: Completing the Square Course: Adv. Alg. & Trig

Completing the Square

Square of BinomialPerfect Square Trinomial

=

x2 bx (b

2)2

(half of b)2

(x b

2)2

The constant (3rd) term of the trinomialis the square of the coefficient of half the trinomial’s x-term.

To make the expression x2 + bx a perfect square, you must add

(1/2 b)2 to the expression.

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Aim: Completing the Square Course: Adv. Alg. & Trig

Solve for x

Find square rootof both sides

Binomial Squared

Add the c term to both sides of equation

Take 1/2 the coefficientof the linear term & square it.

Solving Quadratics by Completing the Square

Complete the square and solve x2 - 6x = 40

x2 - 6x = 40

(6

2)2

+ 9 + 9

(x - 3)2 = 49

(x 3)2 49

x - 3 = ±7

x - 3 = 7 x - 3 = -7x = 10 x = -4

= (-3)2 = 9

Graph this equation

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Aim: Completing the Square Course: Adv. Alg. & Trig

Rewrite the original equation by adding 5

Divide by the equation by a (4)

Binomial squared

Find square rootof both sides

Solve for x

Add 1/16 to each side

Solving Quadratics when a 1

Complete the square and solve 4x2 + 2x - 5 = 0

(x + 1/4)2 = 21/16

x2 + 1/2x = 5/4

(1

4)2

+ 1/16 + 1/16

4x2 + 2x = 5

4x2 + 2x = 54

= x2 + 1/2x = 5/4

1

16

x 1

4

21

16

21

4

x 1

4

21

4Graph this equation

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Aim: Completing the Square Course: Adv. Alg. & Trig

Aim: How do we solve quadratic equations by completing the square?

Find the value of c that makes x2 + 16x + c a perfect square.

Do Now:

Square of BinomialPerfect Square Trinomial

=

x2 bx (b

2)2

(half of b)2

(x b

2)2

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Aim: Completing the Square Course: Adv. Alg. & Trig

(half of 16)2

x2 + 6x = 16a.

Completing the Square Problem 1 - 2

1. Find the value of c that makes x2 + 16x + c a perfect square.

= 64

2. Solve by completing the square.

x2 + 6x + 9 = 16 + 9

(x + 3)2 = 25

x + 3 = ±5

x + 3 = 5 x + 3 = -5x = 2 x = -8

x2 - 4x + 2 = 0b.

x2 - 4x = -2

(x - 2)2 = 2

= (8)2

x2 - 4x + 4 = -2 + 4

x 2 2

x 2 2 x 2 2

x 2 2 x 2 2Graph these equation

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Aim: Completing the Square Course: Adv. Alg. & Trig

Completing the Square Problem 3

Television screens are usually measured by thelength of the diagonal. An oversized televisionhas a 60-inch diagonal. The screen is 12 incheswider than its height. Find the dimensionsof the screen.

SONY

60”

Let x = width of TV x + 12 = length

x2 + (x + 12)2 = 602

x2 + x2 + 24x + 144 = 3600

2x2 + 24x + 144 = 3600

2x2 + 12x + 72 = 1800

Pythagorean theorem

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Aim: Completing the Square Course: Adv. Alg. & Trig

Completing the Square Problem 3 (con’t)

SONY

60”

x2 + 12x + 72 = 1800

x2 + 12x = 1800 - 72

x2 + 12x = 1728

x2 + 12x + 36 = 1728 + 36(x + 6)2 = 1764

(x 6)2 1764

x + 6 = 42

x + 6 = 42 x + 6 = -42

x = 36 x = -48

Width = 36”Length = 36” + 12” = 48”

Graph this equation