1. Section 11.1
Quadratic Equations

2. Quadratic Equations
The general form of the quadratic equation is ax² + bx + c = 0 where a ≠ 0
The unique quadratic equation forms are
ax² + bx + c = 0
ax² + bx = 0
ax² + c = 0 OR ax² - c = 0

3. Zero Product Property Zero Product Rule Theory
If ab = 0 the a = 0 or b = 0
Theory
How can you create a zero through multiplication?
0 * 0
# * 0
0 * #

4. How to solve ax² + bx = 0 Given Factor the Greatest Common Factor
x(ax + b) = 0
Use the zero product property
x = 0 OR ax + b = 0
Solve each equation
x = 0 OR x = -b/a
Check your answers

5. Example Solve 2x² – 3 = 29 Check 2x² = 29 +3 2x² = 32 x² = 16
x = 4 OR x = -4
Check
x = 4 2(4)² – 3 = =29
x = -4 2(-4)² – 3 = =29

6. Example
Solve x² – 1 = 15

7. Example
Solve x² + 5 = 10

8. Square Root Property Square Root Property Theory
For any real number k, if x² = k, the
x = √k or x = -√k
x = ± √k
Theory
Let us factor this problem.
x² – 4 = 0 (x – 2)(x + 2) = 0
x – 2 = 0 OR x + 2 = 0
x = OR x = -2

9. How to solve ax² - c = 0 Given Move the constant to the other side
Divide both sides by the coefficient
x² = c/a
Use the square root property
X = √c/a or -√c/a
Check the solutions

10. Example Solve -5x² = -50 Check -5(x)² = -50 (x)² = 10
x = √10 OR x = -√10
Check
x = √10
-5(√10)² = -5(10) = -50
x = -√10
-5(-√10)² = -5(10) = -50

11. Example
Solve 2x² = 18

12. Example
Solve -x² = 25

13. Example
Solve 2x² = 18

14. How to solve ax² + bx + c = 0
There are three ways to solve this problem
Factoring (5.7)
Completing the square (11.2)
Quadratic Formula (11.2)

15. Completing the square
Completing the square takes the quadratic equation and turns it into a perfect square trinomial.
Then you isolate the variable by using the square root property.

16. Perfect Square Trinomial
x² + 2xb + b² = (x + b)²
x² - 2xb + b² = (x - b)²
We will be creating a perfect square trinomial
We will start with the x² - 2xb
We will create the third term
We will then factor the perfect square trinomial

17. Examples of making the third term
Find the constant for x² + 2x
Find the constant for x² + 4x
Find the constant for x² + 12x
Find the constant for x² – 8x
Find the constant for x² + 21x

18. Generic Third Term
Can you find a pattern to create the third term or new constant?

19. Steps Write equation in decreasing order
Move constant to the other side
Divide both sides by the leading coefficient
Find the new constant
C = [(1/2)b]²
Add the new constant to both sides
Factor one side, simplify the other
Use the square root property
Isolate the variable

20. Solve by Completing the Square
2x² + 8x + 3 = 0
2x² + 8x = -3
x² + 4x = -3/4
New C = [(1/2)(4)]² = 4
x² + 4x + 4 = -3/4 + 4
(x+2)² = 13/4
X+2 = ±√(13/4)
X+2 = ±√13 / 2
X = -2 ±√13 / 2

21. Solve by Completing the Square
X² + 6x = 25

22. Solve by Completing the Square
3X² + 6x = 24

23. Homework
11.1
# 10, 11, 21, 24, 39-50, 51, 53, 55, 57