Factorise and solve x^2 - 8x + 15 = 0.

The easiest method to solve quadratic equations by hand is by factorisation - which means putting the equation into brackets, effectively expressing the equation as a product of two linear expressions. Imagine the brackets are in the form (x+a)(x+b), so when expanded would equal x^2+ax+bx+ab. This shows us that whatever a and b are, they must multiply to make 15 and sum to -8. Now consider the integer factors of 15 (although a and b are not always integers). We have 15 and 1, -15 and -1, 5 and 3, and -5 and -3 (because negative multiplied by negative makes positive). If we sum all the possibilities we can see that -5 and -3 sum to -8, as required, so we therefore have our a and b, a=-3 and b=-5. If you are unsure, substitute these values into the (x+a)(x+b) equation and expand to see if you end up with the same equation. We now have our factorised equation: (x-3)(x-5)=0. It is clear that for two numbers to multiply to make zero, one of the numbers must be zero, so we have the option - either (x-3)=0 or (x-5)=0. We now solve for x, giving x=5 or x=3, as quadratic equations usually have two roots, so this is our final answer.