1. Exploring Quadratic Functions and Inequalities
Advanced Algebra
Chapter 6

2. Solving Quadratic Functions
Solve the following equation.
Square of
a Binomial
Solution:

3. Solving Quadratic Functions
Multiply the following expressions.
Is there a pattern?
Shortcut Method
( x ) = x x
2×product of both terms
1st term
last term
square of
1st term
square of
last term

4. Solving Quadratic Functions
Try using the shortcut method with these.
Now Try Backwards:
x2 + 8x = ( )2
x2 – 4x + 4 = ( )2
x2 + x + ¼ = ( )2
THINK!!!
x + ½

5. Solving Quadratic Functions by Completing the Square
For example, solve the following equation by completing the square.
Step 1  Move the constant to the other side.
Step 2  Notice the coefficient of the linear term is 3, or b = 3. Therefore, is the new constant needed to create a Square Binomial. Add this value to both sides.

6. Solving Quadratic Functions by Completing the Square
Step 3  Factor and Solve.

7. Quadratic Formula
Another way to solve quadratic equations is to use the quadratic formula.
This is derived from the standard form of the equation ax2 + bx + c = 0 by the process of completing the square.

8. Quadratic Formula The Quadratic Formula
The value of the discriminant, b2 – 4ac, determines the nature of the roots of a quadratic equation.
The Discriminant

9. Discriminant  b2 – 4ac Value Description Sample Graph b2 – 4ac
is a perfect
square
b2 – 4ac = 0
b2 – 4ac < 0
b2 – 4ac > 0
Intersects the x-axis once.
One real root.
Does not intersect the x-axis.
Two imaginary roots.
Intersects the x-axis twice.
Two real, irrational roots.
Intersects the x-axis twice.
Two real, rational roots.

10. Solving Quadratic Functions with the Quadratic Formula
For example, solve the following equation with the quadratic formula.
Step 1  Write quadratic equation in Standard Form.
Step 2  Substitute coefficients into quadratic formula. In this case a = 4, b = –20 and c = 25
The discriminant, (–20)2 – 4(4)(25) = 0.
There is one real, rational root.

11. Solving Quadratic Functions with the Quadratic Formula
For example, solve the following equation with the quadratic formula.
Step 1  Write quadratic equation in Standard Form.
Step 2  Substitute coefficients into quadratic formula. In this case a = 3, b = –5 and c = 2
The discriminant, (–5)2 – 4(3)(2) = 1.
There are two real, rational roots.

12. Homework