Factoring quadratic equations

Factoring when a = 1

( x )( x ) = 0.

Now, you need to find two numbers that multiply to give c, and add/subtract to give the coefficient of b. Then you simply put these numbers into the brackets, and sort out the +/- signs correctly.

Example:
x² + x = 12

First, we rearrange into the standard format-

x² - x - 12 = 0

We need two pairs of numbers hat multiply to give 12 and add/subtract to give 1. The numbers are clearly 3 and 4.

Now, write the equation -

(x + 3)(x - 4) = 0

Then, as an essential check, we'd expand the brackets.

Factoring when a does not equal 1

Such as:

3x ² + 7x + 6 = 0

The method is the same is pretty much the same as before. In this case we'd have to find two numbers that multiply with the 3x and x terms in the brackets and then add or subtract to give the value of b. So in this case it'd be 2 in the first bracket and 3 in the second.

Although Noung's writeup shows a good way for a human to factor a trinomial, trying to program an algorithm using that method would be horribly inefficient. It would have to first factor a and c, and then test every possible combination of factors until it finds the correct one. Luckily, there is another method that is much more efficient:

Example: