9.6 THE QUADRATIC FORMULA:

Quadratic Formula: Formula used to find the solution(s) of any quadratic equation.

2 4

2

b b acx

a

If ax2+bx+c = 0 and a ≠ 0,then

GOAL:

QUADRATIC EQUATION:

Ex: What are the solutions to? Use the quadratic formula.

x2-8=2x

SOLUTION: We must first put the equation in ax2+bx+c=0 form.

a = 1 , b = -2 and c = -8 x2-8=2x x2-2x-8 =0

2 4

2

b b acx

a

22 2( ) ( ) 4( )( 8

2( )1

)1x

2 4 32

2x

2 36

2x

2 6

2x

2 6 4

2 22x

2 6 8

24

2x

x = -2, 4

YOU TRY IT:

Ex: What are the solutions to? Use the quadratic formula.

x2-4x=21

SOLUTION: We must first put the equation in ax2+bx+c=0 form.

a = 1 , b = -4 and c = -21 x2-4x=21 x2-4x-21 =0

2 4

2

b b acx

a

24 4( ) ( ) 4( )( )

2

1

)

21

1(x

4 16 84

2x

4 100

2x

4 10

2x

4 10 6

2 23x

4 10 14

2 27x

x = -3, 7

REAL-WORLD:

In the shot put, An athlete throwsA heave metal ballThrough the air. The arc of the ball is modeled by the equation: -0.04x2+.84x+2. Where x is the horizontal distance. How far from the athlete will the ball land?

SOLUTION: The given equation is already in standard form:

a = -0.04 , b = 0.84 and c = 2

-0.04x2+0.84x+2 = 0

2 4

2

b b acx

a

2( ) ( ) 4 00. .04

0.

84 0.8 ( )( )

2( )0

4 2

4x

0.84 1.0256

0.08x

0.84 1.013

0.08x

0.84 1.013

0.0823.16x

x cannot be negative: x = 23.16 meters

0.84 1.013

0.082.16x

VIDEOS: SOLVING BY

FACTORING Solving by factoring:

http://www.khanacademy.org/math/algebra/quadratics/factoring_quadratics/v/Example%201:%20Solving%20a%20quadratic%20equation%20by%20factoring

VIDEOS: SOLVING BY

FACTORING Solving by factoring:

http://www.khanacademy.org/math/trigonometry/polynomial_and_rational/quad_formula_tutorial/v/solving-quadratic-equations-by-completing-the-square

http://www.khanacademy.org/math/algebra/quadratics/factoring_quadratics/v/Example%202:%20Solving%20a%20quadratic%20equation%20by%20factoring

CLASSWORK:

Page 508-509:

Problems: 4, 7, 15, 21, 25, 36, 37.