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