Solve 3m(2m+9)=0

$$3 m ( 2 m + 9 ) = 0$$

Solve for m

$$m = -\frac{9}{2} = -4\frac{1}{2} = -4.5$$

$$m=0$$

Steps Using Factoring

Use the distributive property to multiply 3m by 2m+9.

$$6m^{2}+27m=0$$

Factor out m.

$$m\left(6m+27\right)=0$$

To find equation solutions, solve m=0 and 6m+27=0.

$$m=0 m=-\frac{9}{2}$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 3m by 2m+9.

$$6m^{2}+27m=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 6 for a, 27 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$m=\frac{-27±\sqrt{27^{2}}}{2\times 6}$$

Take the square root of 27^{2}.

$$m=\frac{-27±27}{2\times 6}$$

Multiply 2 times 6.

$$m=\frac{-27±27}{12}$$

Now solve the equation m=\frac{-27±27}{12} when ± is plus. Add -27 to 27.

$$m=\frac{0}{12}$$

Divide 0 by 12.

$$m=0$$

Now solve the equation m=\frac{-27±27}{12} when ± is minus. Subtract 27 from -27.

$$m=-\frac{54}{12}$$

Reduce the fraction \frac{-54}{12} to lowest terms by extracting and canceling out 6.

$$m=-\frac{9}{2}$$

The equation is now solved.

$$m=0 m=-\frac{9}{2}$$

Steps for Completing the Square

Use the distributive property to multiply 3m by 2m+9.

$$6m^{2}+27m=0$$

Divide both sides by 6.

$$\frac{6m^{2}+27m}{6}=\frac{0}{6}$$

Dividing by 6 undoes the multiplication by 6.

$$m^{2}+\frac{27}{6}m=\frac{0}{6}$$

Reduce the fraction \frac{27}{6} to lowest terms by extracting and canceling out 3.

$$m^{2}+\frac{9}{2}m=\frac{0}{6}$$

Divide 0 by 6.

$$m^{2}+\frac{9}{2}m=0$$

Divide \frac{9}{2}, the coefficient of the x term, by 2 to get \frac{9}{4}. Then add the square of \frac{9}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$m^{2}+\frac{9}{2}m+\left(\frac{9}{4}\right)^{2}=\left(\frac{9}{4}\right)^{2}$$

Square \frac{9}{4} by squaring both the numerator and the denominator of the fraction.

$$m^{2}+\frac{9}{2}m+\frac{81}{16}=\frac{81}{16}$$

Factor m^{2}+\frac{9}{2}m+\frac{81}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(m+\frac{9}{4}\right)^{2}=\frac{81}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(m+\frac{9}{4}\right)^{2}}=\sqrt{\frac{81}{16}}$$

Simplify.

$$m+\frac{9}{4}=\frac{9}{4} m+\frac{9}{4}=-\frac{9}{4}$$

Subtract \frac{9}{4} from both sides of the equation.

$$m=0 m=-\frac{9}{2}$$