How do you solve quadratic equations?

A typical GCSE question will give you an equation in the from ax² + bx + c, x² - 8x + 15 = 0 for example, and ask you to solve the possible value of X. To do this we need to split the equation into two halves, also known as factorising, (x + or - one number) * (x + or - another number) = 0. We use x at the start of each of the brackets as the equation is x² and x times x (using the FOIL method) is x². The two numbers we are looking for need to multiply to 15 (the c), and add together to make -8 (the b). There are many ways to find these but the easiest way I find is to first break down the factors of 15, so that would be 1 and 15, 3 and 5, and then the minuses (as a minus times a minus makes a plus) -1 and -15, -3 and -5. Now we look at these and see what adds up to -8. We can see the first one adds up to 16, the second one 8, the third -16 and the final one -8. Now we know the numbers must be -3 and -5 (as they multiply to 15, and add to make -8). We can put them back into the equation above (in any order), so we get (x - 3) * (x - 5) = 0. Now as 0 * anything = 0 for (x - 3) * (x - 5) = 0 to be true one of x - 3 or x - 5 must equal 0, so we know x - 3 = 0 OR x - 5 = 0, from this we can say that x = 3 OR x = 5.