1. Intro: We already know the
standard form of a quadratic
equation is:
y = ax2 + bx + c
The coefficients are: a , b, c
The variables are: y, x
2. The ROOTS (or
solutions) of a
polynomial are
its x-intercepts
The x-intercepts
occur where y =
0.
Roots
3. Example: Find the
roots: y = x2 + x - 6
Solution: Factoring:
y = (x + 3)(x - 2)
0 = (x + 3)(x - 2)
The roots are:
x = -3; x = 2
Roots
4. After centuries of
work,
mathematicians
realized that as long
as you know the
coefficients, you can
find the roots of the
quadratic. Even if it
doesn’t factor!
y ax2
bx c, a 0
x
b b2
4ac
2a
5. 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
7. y 5(1)2
8(1) 3
y 5 8 3
y 0
y 5 3
5
2
8 3
5
3
y 5 9
25
24
5
3
y 45
25
24
5
3
y 9
5
24
5
15
5
y 0
Plug in your
answers for x.
If you’re right,
you’ll get y = 0.
8. 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
4
x
2
4
1
2
x
16
4
4
9. Remember: All the terms must be on one
side BEFORE you use the quadratic
formula.
•Example: Solve 3m2 - 8 = 10m
•Solution: 3m2 - 10m - 8 = 0
•a = 3, b = -10, c = -8
10. Solve: 3x2 = 7 - 2x
Solution: 3x2 + 2x - 7 =
0
a = 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
