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\left(x-3\right)x+1=9\left(x-3\right)
Variable x cannot be equal to 3 since division by zero is not defined. Multiply both sides of the equation by x-3.
x^{2}-3x+1=9\left(x-3\right)
Use the distributive property to multiply x-3 by x.
x^{2}-3x+1=9x-27
Use the distributive property to multiply 9 by x-3.
x^{2}-3x+1-9x=-27
Subtract 9x from both sides.
x^{2}-12x+1=-27
Combine -3x and -9x to get -12x.
x^{2}-12x+1+27=0
Add 27 to both sides.
x^{2}-12x+28=0
Add 1 and 27 to get 28.
x=\frac{-\left(-12\right)±\sqrt{\left(-12\right)^{2}-4\times 28}}{2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -12 for b, and 28 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-12\right)±\sqrt{144-4\times 28}}{2}
Square -12.
x=\frac{-\left(-12\right)±\sqrt{144-112}}{2}
Multiply -4 times 28.
x=\frac{-\left(-12\right)±\sqrt{32}}{2}
Add 144 to -112.
x=\frac{-\left(-12\right)±4\sqrt{2}}{2}
Take the square root of 32.
x=\frac{12±4\sqrt{2}}{2}
The opposite of -12 is 12.
x=\frac{4\sqrt{2}+12}{2}
Now solve the equation x=\frac{12±4\sqrt{2}}{2} when ± is plus. Add 12 to 4\sqrt{2}.
x=2\sqrt{2}+6
Divide 12+4\sqrt{2} by 2.
x=\frac{12-4\sqrt{2}}{2}
Now solve the equation x=\frac{12±4\sqrt{2}}{2} when ± is minus. Subtract 4\sqrt{2} from 12.
x=6-2\sqrt{2}
Divide 12-4\sqrt{2} by 2.
x=2\sqrt{2}+6 x=6-2\sqrt{2}
The equation is now solved.
\left(x-3\right)x+1=9\left(x-3\right)
Variable x cannot be equal to 3 since division by zero is not defined. Multiply both sides of the equation by x-3.
x^{2}-3x+1=9\left(x-3\right)
Use the distributive property to multiply x-3 by x.
x^{2}-3x+1=9x-27
Use the distributive property to multiply 9 by x-3.
x^{2}-3x+1-9x=-27
Subtract 9x from both sides.
x^{2}-12x+1=-27
Combine -3x and -9x to get -12x.
x^{2}-12x=-27-1
Subtract 1 from both sides.
x^{2}-12x=-28
Subtract 1 from -27 to get -28.
x^{2}-12x+\left(-6\right)^{2}=-28+\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=-28+36
Square -6.
x^{2}-12x+36=8
Add -28 to 36.
\left(x-6\right)^{2}=8
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{8}
Take the square root of both sides of the equation.
x-6=2\sqrt{2} x-6=-2\sqrt{2}
Simplify.
x=2\sqrt{2}+6 x=6-2\sqrt{2}
Add 6 to both sides of the equation.