This is all well and good, this "completing the square" method (you might say) but what if the coefficient of x² isn't a perfect square? We can't complete the square in that situation, can we?

Well, not exactly. But there is something we can do. Consider the following:

3x² - 8x - 3 = 0

3 is not a perfect square, but suppose we took this equation and multiplied both sides by 3...then we would have the following:

9x² - 24x -9 = 0

What do we need to do to complete the square now? The constant term needs to be 16, and since it's -16, we need to add 25:

9x² - 24x + 16 = 25
(3x - 4)² = 5²
3x - 4 = 5 or 3x - 4 = -5
3x = 9 or 3x = -1
x = 3 or x = -1/3

Problem #1
Solve for x if 2x² + x - 15 = 0

Solution #1
If we multiply the entire equation by 2, we get:

4x² + 2x - 30 = 0

Thus, to complete the square, the constant term must be 1/4. Since it's -30 (or -120/4), we need to add 121/4 to both sides:

4x² + 2x - 30 + 121/4 = 121/4
4x² + 2x + 1/4 = 121/4
(2x + 1/2)² = (11/2)²
2x + 1/2 = 11/2 or 2x + 1/2 = -11/2
2x = 10/2 or 2x = -12/2
x = 5/2 or x = -3

Problem #2
Solve for x if 12x² - 13x - 4 = 0

Solution #2
You might be tempted to multiply this by 12 to make a perfect square (144), but that's really overkill! Because if we multiply 12 by 3, we get 36, which is also a perfect square. [HINT: in the prime factorization of 12, 3 is the only prime factor which doesn't have an even exponent - that's why I thought to multiply by 3]

So we have 36x² - 39x - 12 = 0

To have a completed square, the constant term must be (13/4)² or 169/16. But it's actually -12, or -192/16. Thus we need to add 361/16:

36x² - 36x -12 + 361/16 = 0 + 361/16
36x² - 36x + 169/16 = 361/16
(6x - 13/4)² = (19/4)²
6x - 13/4 = 19/4 or 6x - 13/4 = -19/4
6x = 32/4 or 6x = -6/4
x = 4/3 or x = -1/4

Special Hint
Sometimes you can make the coefficient of x² a perfect square by dividing instead of multiplying. For example, in 2x² + 2x - 12 = 0, if you notice that all the coefficients are multiples of 2, you can divide out the two, leaving x² + x - 6 = 0, which is much simpler.

For each problem below, solve by completing the square.