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x\left(3x-27\right)=0
Factor out x.
x=0 x=9
To find equation solutions, solve x=0 and 3x-27=0.
3x^{2}-27x=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{-\left(-27\right)±\sqrt{\left(-27\right)^{2}}}{2\times 3}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, -27 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-27\right)±27}{2\times 3}
Take the square root of \left(-27\right)^{2}.
x=\frac{27±27}{2\times 3}
The opposite of -27 is 27.
x=\frac{27±27}{6}
Multiply 2 times 3.
x=\frac{54}{6}
Now solve the equation x=\frac{27±27}{6} when ± is plus. Add 27 to 27.
x=9
Divide 54 by 6.
x=\frac{0}{6}
Now solve the equation x=\frac{27±27}{6} when ± is minus. Subtract 27 from 27.
x=0
Divide 0 by 6.
x=9 x=0
The equation is now solved.
3x^{2}-27x=0
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{3x^{2}-27x}{3}=\frac{0}{3}
Divide both sides by 3.
x^{2}+\left(-\frac{27}{3}\right)x=\frac{0}{3}
Dividing by 3 undoes the multiplication by 3.
x^{2}-9x=\frac{0}{3}
Divide -27 by 3.
x^{2}-9x=0
Divide 0 by 3.
x^{2}-9x+\left(-\frac{9}{2}\right)^{2}=\left(-\frac{9}{2}\right)^{2}
Divide -9, the coefficient of the x term, by 2 to get -\frac{9}{2}. Then add the square of -\frac{9}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}-9x+\frac{81}{4}=\frac{81}{4}
Square -\frac{9}{2} by squaring both the numerator and the denominator of the fraction.
\left(x-\frac{9}{2}\right)^{2}=\frac{81}{4}
Factor x^{2}-9x+\frac{81}{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{9}{2}\right)^{2}}=\sqrt{\frac{81}{4}}
Take the square root of both sides of the equation.
x-\frac{9}{2}=\frac{9}{2} x-\frac{9}{2}=-\frac{9}{2}
Simplify.
x=9 x=0
Add \frac{9}{2} to both sides of the equation.