Question

How do i factor x^2 -5x + 4?

Answer 1

To factor, we want to find a pair of numbers that multiply together to get 4 and add together to get -5.

First, observe that we have to add two numbers together to get -5. We multiply them together to get a positive 4. So, they're both going to be negative to fit these qualities..

We just have to list the negative factors of 4 and check which one adds up to -5:
-1, -4; adds to -5
-2, -2; adds to -4
So, -1 and -4 are the factors: (x - 1)(x - 4).

Check: (x - 1)(x - 4) = x(x - 4) - 1(x - 4)
= x^2 - 4x - x + 4
= x^2 - 5x + 4 yes!

Answer 2

If that process isn't exactly clear, we can see how it makes sense in a general case:
Let x - a and x - b be the factors of the quadratic.
y = (x + a)(x + b) distributing
= x(x + b) + a(x + b) distributing again
= x^2 + bx + ax + ab combine like-terms
= x^2 + (a + b)x + ab <--- the number next to x is the sum of a and b; and the constant at the end is the product of a and b.

Answer 3

So x^2 - 5x + 4 = x^2 + (a + b)x + ab;
ab = 4
a + b = -5
Clearly a=-1 and b=-4 work here. :)