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3x^{2}+13x=0
Swap sides so that all variable terms are on the left hand side.
x\left(3x+13\right)=0
Factor out x.
x=0 x=-\frac{13}{3}
To find equation solutions, solve x=0 and 3x+13=0.
3x^{2}+13x=0
Swap sides so that all variable terms are on the left hand side.
x=\frac{-13±\sqrt{13^{2}}}{2\times 3}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, 13 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-13±13}{2\times 3}
Take the square root of 13^{2}.
x=\frac{-13±13}{6}
Multiply 2 times 3.
x=\frac{0}{6}
Now solve the equation x=\frac{-13±13}{6} when ± is plus. Add -13 to 13.
x=0
Divide 0 by 6.
x=-\frac{26}{6}
Now solve the equation x=\frac{-13±13}{6} when ± is minus. Subtract 13 from -13.
x=-\frac{13}{3}
Reduce the fraction \frac{-26}{6} to lowest terms by extracting and canceling out 2.
x=0 x=-\frac{13}{3}
The equation is now solved.
3x^{2}+13x=0
Swap sides so that all variable terms are on the left hand side.
\frac{3x^{2}+13x}{3}=\frac{0}{3}
Divide both sides by 3.
x^{2}+\frac{13}{3}x=\frac{0}{3}
Dividing by 3 undoes the multiplication by 3.
x^{2}+\frac{13}{3}x=0
Divide 0 by 3.
x^{2}+\frac{13}{3}x+\left(\frac{13}{6}\right)^{2}=\left(\frac{13}{6}\right)^{2}
Divide \frac{13}{3}, the coefficient of the x term, by 2 to get \frac{13}{6}. Then add the square of \frac{13}{6} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}+\frac{13}{3}x+\frac{169}{36}=\frac{169}{36}
Square \frac{13}{6} by squaring both the numerator and the denominator of the fraction.
\left(x+\frac{13}{6}\right)^{2}=\frac{169}{36}
Factor x^{2}+\frac{13}{3}x+\frac{169}{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+\frac{13}{6}\right)^{2}}=\sqrt{\frac{169}{36}}
Take the square root of both sides of the equation.
x+\frac{13}{6}=\frac{13}{6} x+\frac{13}{6}=-\frac{13}{6}
Simplify.
x=0 x=-\frac{13}{3}
Subtract \frac{13}{6} from both sides of the equation.