This document provides instructions for solving quadratic equations by completing the square. It gives examples of solving equations by: 1) Dividing both sides by the leading coefficient, if present 2) Subtracting any constant term not involving the variable 3) Completing the square by adding the square of half the coefficient of the second term 4) Taking the square root of both sides to isolate the variable

1. In this session you will learn to In this lesson you will learn to
•Graph quadratic functions, •Solve quadratic equations by
•Solve quadratic equations. finding the square root.
•Graph exponential functions •Solve quadratic equations by
•Solve problems involving completing the square.
exponential growth and decay.
•Recognize and extend geometric
sequences.
The Gateway Arch in January 2008
Picture: From Wikipedia, the free encyclopedia
http://en.wikipedia.org/wiki/Gateway_Arch
Click to continue.

2. If the trinomial on the left side of the 9 x 2 12 x 4 6
equation is a binomial square,
factor the trinomial, 3x 2 2 6
then take the square root of both sides. 3x 2 2 6
Simplify. 3x 2 6
Solve for the unknown variable by 3x 2 6
adding 2
and dividing by 3.
2 6
x 3

3. If the trinomial on the left side of the 9 x 2 12 x 4 6
equation is a binomial square,
factor the trinomial, 3x 2 2 6
then take the square root of both sides. 3x 2 2 6
Simplify. 3x 2 6
Solve for the unknown variable by 3x 2 6
adding 2
and dividing by 3.
2 6
Click to continue.
x 3

4. If the left side of the equation is not a perfect square trinomial, make the trinomial a
perfect square by using the process of COMPLETING the SQUARE.
2
Solve x 12 x 7 6 by completing the square.
•Remove the third term (7) of the trinomial by subtracting. x 2 12x ___ 1
•Complete the square by adding the square of
x 2
12x 12 2 1 12 2
half the coefficient of the second term. 2 2
•Simplify.
x 2 12 x 36 35
•The trinomial on the left is now a perfect square.
Factor the trinomial on the left.
x 62 35
•Take the square root of both sides.
x 6 35
•Solve for the unknown variable by adding or
subtracting.
x 6 35
Click to continue.

5. If there is a leading coefficient, there is one additional step at the beginning.
Solve 2 x 2 12 x 7 6 by completing the square.
2 x2 12 x 7 6
•The first step is to divide the entire equation by the 2 2 2 2
leading coefficient. Divide by 2. 2 7
x 6x 2
3
•Subtract 7 . x2 6x _ 3 7
2
x2 6x _ 1
2
2
•Complete the square by adding the square of
x 2
6x 62 1 62
half the coefficient of the second term. 2 2 2
•Simplify. x2 6x 32 1
2
32 x 2 6 x 9 17
2
•The trinomial on the left is now a perfect square. x 32 17
2
Factor the trinomial on the left.
•Take the square root of both sides. x 3 17
2
•Solve for the unknown variable by adding or x 3 17
subtracting. 2
Click to continue.

6. In some solutions, there will be fractions. DO NOT be intimated by fractions.
Solve 2 x 2 5x 7 0 by completing the square.
•Divide by 2.
2
2 x2 5
2 x 7
2
0
2
x2 5
2 x 7
2 0
•Subtract 7 . x2 x _ 0 x2 x _
5 7 5 7
2 2 2 2
2
•Complete the square by adding the square of x2 5
x 1 5
2
7 1 5
2
2 2 2 2 2 2
half the coefficient of the second term.
2 2
•Simplify. x2 5
2 x 5
4
7
2
5
4 x2 5
2 x 25
16
7
2
25
16
81
16
•The trinomial on the left is now a perfect square. x 5
2
81
4 16
Factor the trinomial on the left.
•Take the square root of both sides. x 5
4
81
4
9
2
•Solve for the unknown variable by adding or x 5
4
9
4
14
4 , 4
4
subtracting.
Reduce: x 7
2 , 1
Click to end.