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x+11=\left(x-1\right)x+x-1
Variable x cannot be equal to 1 since division by zero is not defined. Multiply both sides of the equation by x-1.
x+11=x^{2}-x+x-1
Use the distributive property to multiply x-1 by x.
x+11=x^{2}-1
Combine -x and x to get 0.
x+11-x^{2}=-1
Subtract x^{2} from both sides.
x+11-x^{2}+1=0
Add 1 to both sides.
x+12-x^{2}=0
Add 11 and 1 to get 12.
-x^{2}+x+12=0
All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.
x=\frac{-1±\sqrt{1^{2}-4\left(-1\right)\times 12}}{2\left(-1\right)}
This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 1 for b, and 12 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-1±\sqrt{1-4\left(-1\right)\times 12}}{2\left(-1\right)}
Square 1.
x=\frac{-1±\sqrt{1+4\times 12}}{2\left(-1\right)}
Multiply -4 times -1.
x=\frac{-1±\sqrt{1+48}}{2\left(-1\right)}
Multiply 4 times 12.
x=\frac{-1±\sqrt{49}}{2\left(-1\right)}
Add 1 to 48.
x=\frac{-1±7}{2\left(-1\right)}
Take the square root of 49.
x=\frac{-1±7}{-2}
Multiply 2 times -1.
x=\frac{6}{-2}
Now solve the equation x=\frac{-1±7}{-2} when ± is plus. Add -1 to 7.
x=-3
Divide 6 by -2.
x=-\frac{8}{-2}
Now solve the equation x=\frac{-1±7}{-2} when ± is minus. Subtract 7 from -1.
x=4
Divide -8 by -2.
x=-3 x=4
The equation is now solved.
x+11=\left(x-1\right)x+x-1
Variable x cannot be equal to 1 since division by zero is not defined. Multiply both sides of the equation by x-1.
x+11=x^{2}-x+x-1
Use the distributive property to multiply x-1 by x.
x+11=x^{2}-1
Combine -x and x to get 0.
x+11-x^{2}=-1
Subtract x^{2} from both sides.
x-x^{2}=-1-11
Subtract 11 from both sides.
x-x^{2}=-12
Subtract 11 from -1 to get -12.
-x^{2}+x=-12
Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.
\frac{-x^{2}+x}{-1}=-\frac{12}{-1}
Divide both sides by -1.
x^{2}+\frac{1}{-1}x=-\frac{12}{-1}
Dividing by -1 undoes the multiplication by -1.
x^{2}-x=-\frac{12}{-1}
Divide 1 by -1.
x^{2}-x=12
Divide -12 by -1.
x^{2}-x+\left(-\frac{1}{2}\right)^{2}=12+\left(-\frac{1}{2}\right)^{2}
Divide -1, the coefficient of the x term, by 2 to get -\frac{1}{2}. Then add the square of -\frac{1}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}-x+\frac{1}{4}=12+\frac{1}{4}
Square -\frac{1}{2} by squaring both the numerator and the denominator of the fraction.
x^{2}-x+\frac{1}{4}=\frac{49}{4}
Add 12 to \frac{1}{4}.
\left(x-\frac{1}{2}\right)^{2}=\frac{49}{4}
Factor x^{2}-x+\frac{1}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.
\sqrt{\left(x-\frac{1}{2}\right)^{2}}=\sqrt{\frac{49}{4}}
Take the square root of both sides of the equation.
x-\frac{1}{2}=\frac{7}{2} x-\frac{1}{2}=-\frac{7}{2}
Simplify.
x=4 x=-3
Add \frac{1}{2} to both sides of the equation.