1. Using the Quadratic Formula to Solve a Quadratic Equation

2. 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.

3. 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”

4. Solving a Quadratic with the Quadratic Formula
Algebraically solve:
Must equal 0
Find the values of “a,” “b,” “c”
a = b = c = 1 -3 -7
Simplify the expression in the square root first
Since the square root can not simplified, this is an acceptable EXACT answer
Substitute into the Quadratic Formula
Or you can approximate the expressions (don’t forget parentheses). This is NOT exact.
Or you can write two expressions. One with addition in the numerator and other with subtraction. This is also an EXACT answer.

5. Solving a Quadratic with the Quadratic Formula
Algebraically solve:
Must equal 0
Find the values of “a,” “b,” “c”
a = b = c = 2 6 -5
Simplify the expression in the square root first
The square root can be simplified.
Substitute into the Quadratic Formula
The GCF of every term is 2

6. Solving a Quadratic: Make Sure to Isolate 0
Solve:
Find the values of “a,” “b,” “c”
Solve for 0 first!
a = b = c = 1 -3 -4
Distribute
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.

7. Solving a Quadratic with the Quadratic Formula: Two Solutions
Algebraically solve:
Must equal 0
Find the values of “a,” “b,” “c”
a = b = c = 4
-121
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.

8. Solving a Quadratic with the Quadratic Formula: No Solutions
Algebraically solve:
Must equal 0
Find the values of “a,” “b,” “c”
a = b = c = 1 -5 9
Simplify the expression in the square root first
Substitute into the Quadratic Formula
This can not be calculated because you can not square root a negative.
The graph of the quadratic has no x-intercepts.
NO SOLUTIONS

9. Solving a Quadratic with the Quadratic Formula: One Solution
Algebraically solve:
Must equal 0
Find the values of “a,” “b,” “c”
a = b = c = 36
-60 25
Simplify the expression in the square root first
The square root of 0 is 0.
Substitute into the Quadratic Formula
The is no difference to adding zero or subtracting 0. This expression will result in only one answer.

10. The Discriminant of a Quadratic
For ANY 0 = ax2 + bx +c (standard form) the value given by:
If the Discriminant (the value underneath the square root in the quadratic formula) is….
Greater than zero there are two roots
Equal to zero there is one root
Less than zero there are no roots