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x^{2}-6x=9
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}-6x-9=9-9
Subtract 9 from both sides of the equation.
x^{2}-6x-9=0
Subtracting 9 from itself leaves 0.
x=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\left(-9\right)}}{2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -6 for b, and -9 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-6\right)±\sqrt{36-4\left(-9\right)}}{2}
Square -6.
x=\frac{-\left(-6\right)±\sqrt{36+36}}{2}
Multiply -4 times -9.
x=\frac{-\left(-6\right)±\sqrt{72}}{2}
Add 36 to 36.
x=\frac{-\left(-6\right)±6\sqrt{2}}{2}
Take the square root of 72.
x=\frac{6±6\sqrt{2}}{2}
The opposite of -6 is 6.
x=\frac{6\sqrt{2}+6}{2}
Now solve the equation x=\frac{6±6\sqrt{2}}{2} when ± is plus. Add 6 to 6\sqrt{2}.
x=3\sqrt{2}+3
Divide 6+6\sqrt{2} by 2.
x=\frac{6-6\sqrt{2}}{2}
Now solve the equation x=\frac{6±6\sqrt{2}}{2} when ± is minus. Subtract 6\sqrt{2} from 6.
x=3-3\sqrt{2}
Divide 6-6\sqrt{2} by 2.
x=3\sqrt{2}+3 x=3-3\sqrt{2}
The equation is now solved.
x^{2}-6x=9
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}-6x+\left(-3\right)^{2}=9+\left(-3\right)^{2}
Divide -6, the coefficient of the x term, by 2 to get -3. Then add the square of -3 to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}-6x+9=9+9
Square -3.
x^{2}-6x+9=18
Add 9 to 9.
\left(x-3\right)^{2}=18
Factor x^{2}-6x+9. 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-3\right)^{2}}=\sqrt{18}
Take the square root of both sides of the equation.
x-3=3\sqrt{2} x-3=-3\sqrt{2}
Simplify.
x=3\sqrt{2}+3 x=3-3\sqrt{2}
Add 3 to both sides of the equation.