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x^{2}-12x=11
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^{2}-12x-11=11-11
Subtract 11 from both sides of the equation.
x^{2}-12x-11=0
Subtracting 11 from itself leaves 0.
x=\frac{-\left(-12\right)±\sqrt{\left(-12\right)^{2}-4\left(-11\right)}}{2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -12 for b, and -11 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-12\right)±\sqrt{144-4\left(-11\right)}}{2}
Square -12.
x=\frac{-\left(-12\right)±\sqrt{144+44}}{2}
Multiply -4 times -11.
x=\frac{-\left(-12\right)±\sqrt{188}}{2}
Add 144 to 44.
x=\frac{-\left(-12\right)±2\sqrt{47}}{2}
Take the square root of 188.
x=\frac{12±2\sqrt{47}}{2}
The opposite of -12 is 12.
x=\frac{2\sqrt{47}+12}{2}
Now solve the equation x=\frac{12±2\sqrt{47}}{2} when ± is plus. Add 12 to 2\sqrt{47}.
x=\sqrt{47}+6
Divide 12+2\sqrt{47} by 2.
x=\frac{12-2\sqrt{47}}{2}
Now solve the equation x=\frac{12±2\sqrt{47}}{2} when ± is minus. Subtract 2\sqrt{47} from 12.
x=6-\sqrt{47}
Divide 12-2\sqrt{47} by 2.
x=\sqrt{47}+6 x=6-\sqrt{47}
The equation is now solved.
x^{2}-12x=11
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.
x^{2}-12x+\left(-6\right)^{2}=11+\left(-6\right)^{2}
Divide -12, the coefficient of the x term, by 2 to get -6. Then add the square of -6 to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}-12x+36=11+36
Square -6.
x^{2}-12x+36=47
Add 11 to 36.
\left(x-6\right)^{2}=47
Factor x^{2}-12x+36. 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-6\right)^{2}}=\sqrt{47}
Take the square root of both sides of the equation.
x-6=\sqrt{47} x-6=-\sqrt{47}
Simplify.
x=\sqrt{47}+6 x=6-\sqrt{47}
Add 6 to both sides of the equation.