quadratic equations
DESCRIPTION
SchoolTRANSCRIPT
•Intro:Intro: We already know the We already know the standard form of a quadratic standard form of a quadratic equation is: equation is:
y = y = aaxx22 + + bbx + x + cc•The The constantsconstants are: are: a , b, ca , b, c•The The variablesvariables are: are: y, xy, x
•The The ROOTSROOTS (or (or solutionssolutions) of a ) of a polynomial are polynomial are its its x-interceptsx-intercepts
•Recall: The Recall: The x-x-interceptsintercepts occur where occur where y y = 0= 0..
Roots
•Example:Example: Find Find the roots: the roots: y = xy = x22 + x - 6+ x - 6
•Solution:Solution: Factoring: Factoring: y = (x y = (x + 3)(x - 2)+ 3)(x - 2)
•0 = (x + 3)(x - 2)0 = (x + 3)(x - 2)•The roots are: The roots are: •x = -3; x = 2x = -3; x = 2
Roots
•After centuries of After centuries of work, work, mathematicians mathematicians realized that as realized that as long as you know long as you know the coefficients, the coefficients, you can find the you can find the roots of the roots of the quadratic. Even if quadratic. Even if it doesn’t factor!it doesn’t factor!
y ax2 bx c, a 0
x b b2 4ac
2a
Solve: y = 5x2 8x 3
x b b2 4ac
2a
a 5, b 8, c 3
x ( 8) ( 8)2 4(5)(3)
2(5)
x 8 64 60
10
x 8 4
10
x 8 2
10
x 8 2
10
x 8 2
10
10
101
x 8 2
10
6
10
3
5Roots
y 5(1)2 8(1) 3y 5 8 3y 0
y 5 35 2
8 35 3
y 5 925 24
5 3
y 4525 24
5 3
y 95 24
5 155
y 0
Plug in your Plug in your answers for answers for xx..If you’re right, If you’re right, you’ll get you’ll get y = y = 00..
Solve : y 2x2 7x 4
a 2, b 7, c 4
x b b2 4ac
2a
x (7) (7)2 4(2)( 4)
2(2)
x 7 49 32
4
x 7 81
4
x 7 9
4x
2
4
1
2
x 16
4 4
RememberRemember: All the terms must : All the terms must be on be on one sideone side BEFORE you use BEFORE you use the quadratic formula.the quadratic formula.
•Example:Example: Solve Solve 3m3m22 - 8 = 10m - 8 = 10m
•Solution:Solution: 3m3m2 2 - 10m - 8 = 0- 10m - 8 = 0
•a = 3, b = -10, c = -8a = 3, b = -10, c = -8
•Solve:Solve: 3x 3x22 = 7 - 2x = 7 - 2x•Solution:Solution: 3x 3x22 + 2x - 7 = 0 + 2x - 7 = 0•a = 3, b = 2, c = -7a = 3, b = 2, c = -7
x b b2 4ac
2a
x (2) (2)2 4(3)( 7)
2(3)
x 2 4 84
6
x 2 88
6
x 2 4 • 22
6
x 2 2 22
6 x 1 22
3