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Solving Quadratics by Completing the Square

Reference > Mathematics > Algebra > The Quadratic Formula
 

In the last section we explored how to find out what we would need to add to a quadratic to turn it into a perfect square. But if you understand algebra, you understand that you can't just arbitrarily add something to an expression without changing its value.

However, if the expression is part of an equation, then you can add the same thing to both sides of the equation, without altering the meaning of the equation.

Why is this useful?

Consider the following:

x2 + 4x + 3 = 0

What would you need to add to the left side of the equation to make it a perfect square? From the last section you know that you would need to add 1. But if you add 1 to the left side, you also have to add it to the right side:

x2 + 4x + 3 + 1 = 0 + 1
x2 + 4x + 4 = 1

Both sides of this equation are perfect squares:

(x + 2)2 = 12

Thus, either x + 2 = 1 or x + 2 = -1. 

This leads to x = -1 or x = -3, and we've solved the equation by completing the square!

Problem #1
Solve for x: x2 + 8x + 12 = 0

Solution #1
To make the left hand side of the equation a perfect square, we would have to add 4. This leads to:

x2 + 8x + 12 + 4 = 0 + 4
x2 + 8x + 16 = 4
(x + 4)2 = 22

Thus, x + 4 = 2 or x + 4 = -2, leading to:

x = -2 or x = -6

Problem #2
Solve for x: 4x2 + 36x + 65 = 0

Solution #2
In order to make the left hand side a perfect square, we would need to add 16.

4x2 + 36x + 65 +16 = 0 + 16
4x2 + 36x + 81 = 16
(2x + 9)2=42
2x + 9 = 4 or 2x + 9 = -4
x = -5/2 or x = -13/2

For each problem below, complete the square to solve the quadratic.

Problem #3
Solve for x: 9x2 + 9x + 2 = 0

Solution #3
This one is a little messier, because a fraction is required to make a perfect square. The constant term has to be (3/2)2 = 9/4. Since it's actually 2 (or 8/4), we need to add 1/4 to make a perfect square:

9x2 + 9x + 9/4 = 1/4
(3x + 3/2)2 = (1/2)2
3x + 3/2 = 1/2 or 3x + 3/2 = -1/2
3x = -1 or 3x = -2
x = -1/3 or x = -2/3

Questions

1.
x2 + 6x + 8 = 0
2.
x2 - 10x + 9 = 0
3.
4x2 - 2x - 12 = 0
4.
x2 + 2x = 0
5.
81x2 - 18x - 80 = 0
6.
9x2 - 78x + 168 = 0
7.
9x2 + 9x + 2 = 0
8.
25x2 + 15x - 28 = 0
9.
100x2 - 40x + 3 = 0
10.
16x2 + 48x + 35 = 0
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Completing a SquareCompleting a Square
Coefficient of x SquaredCoefficient of x Squared
 

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