First, add 1111 to both sides of the equation
x^2-10x+11=-11+11x2−10x+11=−11+11
x^2-10x+11 = 0x2−10x+11=0
There are no nice integers which multiply together to make +11+11 and add together to make -10−10. 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(x2−10x+ )+11+ =0
Take the coefficient of xx, in this case -10−10, divide it by 22 and square the final result.
((-10)/2)^2=25(−102)2=25
In the spaces provided, first add 2525, then subtract it, as follows:
(x^2-10x+25)+11-25=0(x2−10x+25)+11−25=0
This makes a perfect square inside the parenthesis
(x-5)^2+11-25=0(x−5)2+11−25=0
(x-5)^2-14=0(x−5)2−14=0
(x-5)^2=14(x−5)2=14
Take the square root of both sides
x-5=+-sqrt(14)x−5=±√14
x=5+-sqrt(14)x=5±√14
This gives two results
x=5+sqrt(14)x=5+√14 and x=5-sqrt(14)x=5−√14
Solving both for zero gives your factors
x-5-sqrt(14)=0x−5−√14=0 and x-5+sqrt(14)=0x−5+√14=0
Thus, the original equation factors as follows:
x^2-10x+11=(x-5-sqrt(14))(x-5+sqrt(14))x2−10x+11=(x−5−√14)(x−5+√14)
