1. MTH 065 Elementary Algebra II
Chapter 11
Quadratic Functions and Equations
Section 11.1
Quadratic Equations

2. Geometric Representation of Completing the Squarex
x + 8
Area = x(x + 8)

3. Geometric Representation of Completing the Squarex x2 8x x 8
Area = x2 + 8x

4. Geometric Representation of Completing the Squarex x2 8x x 8
Area = x2 + 8x

5. Geometric Representation of Completing the Squarex x2 8x x 4 4
Area = x2 + 8x

6. Geometric Representation of Completing the Squarex x2 4x 4x x 4 4
Area = x2 + 8x

7. Geometric Representation of Completing the Squarex x2 4x 4x x 4 4
Area = x2 + 8x

8. Geometric Representation of Completing the Square4 4x x x2 4x x 4
Area = x2 + 8x

9. Geometric Representation of Completing the Square4 4x ? x x2 4x x 4
Area = x2 + 8x + ?

10. Geometric Representation of Completing the Square4 4x 16 x x2 4x x 4
Area = x2 + 8x + 16

11. Geometric Representation of Completing the Square4 4x 16 x x2 4x x 4
Area = x2 + 8x + 16 = (x + 4)2

12. Terminology ax2 + bx + c = 0 f(x) = ax2 + bx + c Quadratic Equation
Any equation equivalent to an equation with the form …
ax2 + bx + c = 0
… where a, b, & c are constants and a ≠ 0.
Quadratic Function
Any function equivalent to the form …
f(x) = ax2 + bx + c
... where a, b, & c are constants and a ≠ 0.

13. Review Results from Chapter 6
Solve quadratic equations by graphing.
Put into standard form: ax2 + bx + c = 0
Graph the function: f(x) = ax2 + bx + c
Solutions are the x-intercepts.
# of Solutions? , 1, or 2
Details of Graphs of Quadratic Functions – Section 11.6

14. Review Results from Chapter 6
Solve quadratic equations by factoring.
Put into standard form: ax2 + bx + c = 0
Factor the quadratic: (rx + m)(sx + n) = 0
Set each factor equal to zero and solve.
# of Solutions?
0  does not factor (not factorable  no solution)
1  factors as a perfect square (if it factors)
2  two different factors (if it factors)

15. Principle of Square Roots
For any number k, if … … then …

16. Principle of Square Roots
For any number k, if … … then …
Why? Consider the following example …
x2 = 9
 x2 – 9 = 0
 (x – 3)(x + 3) = 0
 x = 3, –3

17. Application of the Principle of Square Roots
Solve the equation …
Note
This example demonstrates how to solve a quadratic equation with no linear (bx) term.

18. Application of the Principle of Square Roots
Solve the equation …
Note
Remember to always simplify radicals.
no perfect squares
no multiples of perfect squares
no negatives

19. Application of the Principle of Square Roots
Solve the equations …

20. Application of the Principle of Square Roots
Solve the equation …
But this does not factor …

21. Solving by “Completing the Square”
Note: This polynomial does not factor.

22. Solving ax2 + bx + c = 0 by “Completing the Square”
Basic Steps …
Get into the form: ax2 + bx = d
Divide through by a giving: x2 + mx = n
Add the square of half of m to both sides.
i.e. add
Factor the left side (a perfect square).
Solve using the Principle of Square Roots.