Solve for x (complex solution)
x=\frac{1+\sqrt{55}i}{12}\approx 0.083333333+0.618016541i
x=\frac{-\sqrt{55}i+1}{12}\approx 0.083333333-0.618016541i
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9x^{2}-6x+1+18=\left(3-3x\right)\left(3x+4\right)
Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(3x-1\right)^{2}.
9x^{2}-6x+19=\left(3-3x\right)\left(3x+4\right)
Add 1 and 18 to get 19.
9x^{2}-6x+19=-3x+12-9x^{2}
Use the distributive property to multiply 3-3x by 3x+4 and combine like terms.
9x^{2}-6x+19+3x=12-9x^{2}
Add 3x to both sides.
9x^{2}-3x+19=12-9x^{2}
Combine -6x and 3x to get -3x.
9x^{2}-3x+19-12=-9x^{2}
Subtract 12 from both sides.
9x^{2}-3x+7=-9x^{2}
Subtract 12 from 19 to get 7.
9x^{2}-3x+7+9x^{2}=0
Add 9x^{2} to both sides.
18x^{2}-3x+7=0
Combine 9x^{2} and 9x^{2} to get 18x^{2}.
x=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\times 18\times 7}}{2\times 18}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 18 for a, -3 for b, and 7 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-3\right)±\sqrt{9-4\times 18\times 7}}{2\times 18}
Square -3.
x=\frac{-\left(-3\right)±\sqrt{9-72\times 7}}{2\times 18}
Multiply -4 times 18.
x=\frac{-\left(-3\right)±\sqrt{9-504}}{2\times 18}
Multiply -72 times 7.
x=\frac{-\left(-3\right)±\sqrt{-495}}{2\times 18}
Add 9 to -504.
x=\frac{-\left(-3\right)±3\sqrt{55}i}{2\times 18}
Take the square root of -495.
x=\frac{3±3\sqrt{55}i}{2\times 18}
The opposite of -3 is 3.
x=\frac{3±3\sqrt{55}i}{36}
Multiply 2 times 18.
x=\frac{3+3\sqrt{55}i}{36}
Now solve the equation x=\frac{3±3\sqrt{55}i}{36} when ± is plus. Add 3 to 3i\sqrt{55}.
x=\frac{1+\sqrt{55}i}{12}
Divide 3+3i\sqrt{55} by 36.
x=\frac{-3\sqrt{55}i+3}{36}
Now solve the equation x=\frac{3±3\sqrt{55}i}{36} when ± is minus. Subtract 3i\sqrt{55} from 3.
x=\frac{-\sqrt{55}i+1}{12}
Divide 3-3i\sqrt{55} by 36.
x=\frac{1+\sqrt{55}i}{12} x=\frac{-\sqrt{55}i+1}{12}
The equation is now solved.
9x^{2}-6x+1+18=\left(3-3x\right)\left(3x+4\right)
Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(3x-1\right)^{2}.
9x^{2}-6x+19=\left(3-3x\right)\left(3x+4\right)
Add 1 and 18 to get 19.
9x^{2}-6x+19=-3x+12-9x^{2}
Use the distributive property to multiply 3-3x by 3x+4 and combine like terms.
9x^{2}-6x+19+3x=12-9x^{2}
Add 3x to both sides.
9x^{2}-3x+19=12-9x^{2}
Combine -6x and 3x to get -3x.
9x^{2}-3x+19+9x^{2}=12
Add 9x^{2} to both sides.
18x^{2}-3x+19=12
Combine 9x^{2} and 9x^{2} to get 18x^{2}.
18x^{2}-3x=12-19
Subtract 19 from both sides.
18x^{2}-3x=-7
Subtract 19 from 12 to get -7.
\frac{18x^{2}-3x}{18}=-\frac{7}{18}
Divide both sides by 18.
x^{2}+\left(-\frac{3}{18}\right)x=-\frac{7}{18}
Dividing by 18 undoes the multiplication by 18.
x^{2}-\frac{1}{6}x=-\frac{7}{18}
Reduce the fraction \frac{-3}{18} to lowest terms by extracting and canceling out 3.
x^{2}-\frac{1}{6}x+\left(-\frac{1}{12}\right)^{2}=-\frac{7}{18}+\left(-\frac{1}{12}\right)^{2}
Divide -\frac{1}{6}, the coefficient of the x term, by 2 to get -\frac{1}{12}. Then add the square of -\frac{1}{12} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}-\frac{1}{6}x+\frac{1}{144}=-\frac{7}{18}+\frac{1}{144}
Square -\frac{1}{12} by squaring both the numerator and the denominator of the fraction.
x^{2}-\frac{1}{6}x+\frac{1}{144}=-\frac{55}{144}
Add -\frac{7}{18} to \frac{1}{144} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
\left(x-\frac{1}{12}\right)^{2}=-\frac{55}{144}
Factor x^{2}-\frac{1}{6}x+\frac{1}{144}. 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}{12}\right)^{2}}=\sqrt{-\frac{55}{144}}
Take the square root of both sides of the equation.
x-\frac{1}{12}=\frac{\sqrt{55}i}{12} x-\frac{1}{12}=-\frac{\sqrt{55}i}{12}
Simplify.
x=\frac{1+\sqrt{55}i}{12} x=\frac{-\sqrt{55}i+1}{12}
Add \frac{1}{12} to both sides of the equation.
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