Factorize x^2-x-20
Let's think backwards - what can we multiply to get this result?First, let's take a simpler example, an expression:(x-1)(x-2). We multiply 1st term of the 1st brackets by terms of the 2nd bracket and then we do the same with the second term of the 1st bracket: (x-1)(x-2)=xx+x(-2)+(-1)x+(-1)(-2)=x2-2x-x+2=x2-3x+2 When simplified, we also obtain a 3-term quadratic expression, similar to the one given in the question.The last coefficient is obtained through multiplying the second terms of both brackets: (-1)*(-2)=2It is the coefficient of x that is more problematic one: it is in fact a sum of two coefficients: -2 and -1, which are the second terms of two brackets. We can simply say that they add up to give a coefficient of x in the quadratic expression.Let's come back to the example we investigate. Say our factorized expression looks like (x-a)(x-b) and we are looking for a and b. Let's look for the integers that multiply to give -20 and add up to give -1.For the first criterion there are many choices, -10 and 2, -4 and 5, -5 and 4, -20 and 1 and so on. But only one pair from these adds up to give -1: -5 and 4.So our expression is (x-(-5))(x-4), which is (x+5)(x-4).In fact our a and b could also be 4 and -5, respectively. Substitute them and you will see this gives the same result, except for brackets being swapped, which does not make a difference since multiplication is commutative - can be done in either way.The answer is: (x+5)(x-4)
