1. Finding the Solutions, x-intercepts, Roots, or Zeros of A Quadratic

2. x–Intercepts, Solutions, Roots, and Zeros in Quadratics
x-intercept(s): Where the graph of y = ax2 + bx + c crosses the x-axis. The value(s) for x that makes a quadratic equal 0.
Solution(s) OR Roots: The value(s) of x that satisfies 0 = ax2 + bx + c.
Zeros: The value(s) of x that make ax2 + bx + c equal 0.

3. If a . b = 0, then a and or b is equal to 0
Zero Product Property
If a . b = 0, then a and or b is equal to 0
Ex: Solve the following equation below.
0 = ( x + 14 )( 6x + 1 )
Would you rather solve the equation above or this: 0 = 6x2 + 85x + 14 ?

4. Use the Zero-Product Property
Example
Solve:
Factor to rewrite as a product
Product c
(2x2)(-12)
420x2 5
Solve for 0 first!
30x
14x 35
ax2c
12x2 2x
30x x
GCF
___
ax2 bx 6x 7
44x
Use the Zero-Product Property
Sum

5. But this parabola has two zeros.
Example
Use the Zero Product Property to find the roots of:
Product
But this parabola has two zeros.
(x2)(-7)
-7x2 c -7
ax2c
IMPOSSIBLE x2 bx
___
ax2
-3x
Sum
Just because a quadratic is not factorable, does not mean it does not have roots. Thus, there is a need for a new algebraic method to find these roots.

6. Quadratic Formula
For ANY 0 = ax2 + bx +c (standard form) the value(s) of x is given by:
MUST equal 0
Plus or Minus
Opposite of b
“All Over”
This formula will provide the solutions (or lack thereof) to ANY Quadratic.

7. Example a = b = c = 1 -3 -4 Solve: Find the values of “a,” “b,” “c”
Solve for 0 first!
a = b = c = 1 -3 -4
Simplify the expression in the square root first
The square root can be simplified.
Substitute into the Quadratic Formula Or
Since the answers will be rational, it is best to list both.