How do you factor x ^ { 2} - 10x = - 11?

1 Answer
Andy Y.
Nov 9, 2017

x^2-10x+11=(x-5-sqrt(14))(x-5+sqrt(14))

Explanation:

First, add 11 to both sides of the equation

x^2-10x+11=-11+11

x^2-10x+11 = 0

There are no nice integers which multiply together to make +11 and add together to make -10. So factoring will involve using either the quadratic equation or completing the square.

Let's choose completing the square. Rewrite the above equation with spaces to prepare for the next step.

(x^2-10x+" ")+11 + " "=0

Take the coefficient of x, in this case -10, divide it by 2 and square the final result.

((-10)/2)^2=25

In the spaces provided, first add 25, then subtract it, as follows:

(x^2-10x+25)+11-25=0

This makes a perfect square inside the parenthesis

(x-5)^2+11-25=0

(x-5)^2-14=0

(x-5)^2=14

Take the square root of both sides

x-5=+-sqrt(14)

x=5+-sqrt(14)

This gives two results

x=5+sqrt(14) and x=5-sqrt(14)

Solving both for zero gives your factors

x-5-sqrt(14)=0 and x-5+sqrt(14)=0

Thus, the original equation factors as follows:

x^2-10x+11=(x-5-sqrt(14))(x-5+sqrt(14))

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