Question

How do you factor the expression `x^2 + 2x + 56`?

Answer 1

`x^2+2x+56 = (x+1-sqrt(55)i)(x+1+sqrt(55)i)`

Explanation:

`x^2+2x+56` is in the form `ax^2+bx+c`, with `a=1`, `b=2` and `c=56`

This has discriminant `Delta` given by the formula:

`Delta = b^2-4ac = 2^2-4(1)(56) = 4-224 = -220`

Since `Delta < 0` this quadratic has no Real zeros and no linear factors with Real coefficients.

We can factor it with the aid of Complex coefficients by completing the square as follows:

`x^2+2x+56 = x^2+2x+1+55`

`color(white)(x^2+2x+56) = (x+1)^2-(sqrt(55)i)^2`

`color(white)(x^2+2x+56) = ((x+1)-sqrt(55)i)((x+1)+sqrt(55)i)`

`color(white)(x^2+2x+56) = (x+1-sqrt(55)i)(x+1+sqrt(55)i)`