Quadratics and RootsQuadratics and Roots

By Petrain KingIMaST Lead CoachLAUSD

Modified from a PowerPoint by Mark P of the same title

http://subjectsearch.wikispaces.com/Math

Chapter 13-4 of Prentice HallChapter 13-4 of Prentice Hall

• What are quadratic What are quadratic equations?equations?

• Solving Quadratic Equations Solving Quadratic Equations for ROOTS.for ROOTS.

• How many solutions?How many solutions?

What are quadratic equations?What are quadratic equations?

• Any equation of the form:Any equation of the form:y=axy=ax22+bx + c+bx + c

• The highest power of the variable is: The highest power of the variable is: 22

RootsRoots

Where are they in this example?

RootsRoots

Where are they in this example?

X= Time

RootsRoots

Where are they in this example?

X= Time

Y= Height

RootRoot

What do quadratic equations look What do quadratic equations look like?like?

• The name for the graph of quadratics is a:The name for the graph of quadratics is a:parabolaparabola

• If the xIf the x22 term is term is positivepositive the “bowl” opens : the “bowl” opens :upwardupward

• If the xIf the x22 term is negative the “bowl” opens: term is negative the “bowl” opens:downwarddownward

What do quadratic equations look like?What do quadratic equations look like?

If the If the xx22 term is term is positivepositive

If the If the xx22 term is term is negativenegative

Example One; Page 590Example One; Page 590

5x5x22-8x= -3-8x= -3

5x5x22-8x+3=0-8x+3=0

5x5x22-8x= -3-8x= -35x5x22-8x-8x == 33

+3+3 +3 +3-3+3=0

5x5x22-8x+3=0-8x+3=0

• AA• BB• CC

5-83

5x5x22-8x+3=0-8x+3=0AA BB C C

5x5x22-8x+3=0-8x+3=0AA B CB C

-b±√b-b±√b22-4ac-4ac2a

5x5x22-8x+3=0 -8x+3=0 aa b b cc

-b±√b-b±√b22-4ac-4ac2a

-(-)-(-)88±√-±√-8822-4(-4(55)()(33))2(5)

5x5x22-8x+3=0 -8x+3=0 aa b b cc

-b±√b-b±√b22-4ac-4ac2a

-(-)-(-)88±√±√8822-4(-4(55)()(33))2(5)

Be careful Be very careful

88±√-±√-8822-4(-4(55)()(33))2(5)

88±√-±√-8822-4(-4(1515))10

88±√64±√64 - 60- 6010

88±√±√8822-4(-4(55)()(33))2(5)

88±√64±√64 - 60- 6010

The given 4 wasmultiplied with a

and cThe given 2 wasmultiplied with a

88±√64±√64 - 60- 6010

88±√4±√4

1010

The difference between-60 and +64

88±√4±√410

8 8 ±± 22

1010

What’s the square root of 4?

10

0.8 ± 0.2± 0.2

0.8 ± 0.2± 0.20.8 + 0.2 = 1.00.8 + 0.2 = 1.0

0.8 - 0.2 = 0.8 - 0.2 = 0.600.60

The Solution ARE1 and 3/5

6/10 = 3/5= 0.6

Quiz TimeQuiz Time

A.A.2x2x22 = 4-7x = 4-7x

B.B. 3x3x22 - 8 = 10x - 8 = 10x

HomeworkHomework

• Page 593 Page 593

– Problems 1-3, 7-12, 15-18Problems 1-3, 7-12, 15-18

Using Quadratic Equations.Using Quadratic Equations.One exampleOne example

• The path of a baseball thrown into the air The path of a baseball thrown into the air can be described by this quadratic:can be described by this quadratic:– h = -16xh = -16x22 + 10x + 3 (h=height, t=time) + 10x + 3 (h=height, t=time)

• Using this equation, we can find the height Using this equation, we can find the height of the ball after any amount of time by of the ball after any amount of time by substituting a “t” value into the equation substituting a “t” value into the equation and solving.and solving.

Solving Quadratic Equations.Solving Quadratic Equations.

• To solve the quadratic equation for x we To solve the quadratic equation for x we must use the Quadratic Formula. Have you must use the Quadratic Formula. Have you memorized it yet?memorized it yet?

– x = -b ± bx = -b ± b22 - 4ac - 4ac 2a 2a

How many solutions?How many solutions?

• A quick way to find out how many A quick way to find out how many solutions a quadratic has, simply find the solutions a quadratic has, simply find the value of the discriminent.value of the discriminent.

– If bIf b22-4ac > 0 the are 2 solutions-4ac > 0 the are 2 solutions– If bIf b22-4ac = 0 there is only 1 solution-4ac = 0 there is only 1 solution– If bIf b22-4ac < 0 there are no solutions. Why?-4ac < 0 there are no solutions. Why?

• We can’t evaluate the square root of a negative number.