How do you factor the expression x^2-x-36?

1 Answer
George C.
Jan 5, 2016

Use the quadratic formula to find:

x^2-x-36 = (x-(1-sqrt(145))/2)(x-(1+sqrt(145))/2)

Explanation:

You would like to find a pair of factors of 36 that differ by 1. Unfortunately, no such pair exists, so this polynomial will not factor with integer coefficients.

Let's check the discriminant: x^2-x-36 is in the form ax+bx+c with a=1, b=-1 and c=-36. This has discriminant Delta given by the formula:

Delta = b^2-4ac = (-1)^2-(4*1*-36) = 1+144 = 145

Since this is positive but not a perfect square, the quadratic has factors with Real irrational coefficients.

We can find these factors using the quadratic formula. The zeros of the quadratic are given by:

x = (-b+-sqrt(b^2-4ac))/(2a) = (-b+-sqrt(Delta))/(2a)

=(1+-sqrt(145))/2

That is x_1 = (1-sqrt(145))/2 and x_2 = (1+sqrt(145))/2

So:

x^2-x-36 = (x-x_1)(x-x_2) = (x-(1-sqrt(145))/2)(x-(1+sqrt(145))/2)

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