Solve 30r^2=-15r

$$30 r ^ { 2 } = - 15 r$$

Solve for r

$$r=-\frac{1}{2}=-0.5$$

$$r=0$$

Steps Using Factoring

Add 15r to both sides.

$$30r^{2}+15r=0$$

Factor out r.

$$r\left(30r+15\right)=0$$

To find equation solutions, solve r=0 and 30r+15=0.

$$r=0 r=-\frac{1}{2}$$

Steps Using the Quadratic Formula

Add 15r to both sides.

$$30r^{2}+15r=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 30 for a, 15 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$r=\frac{-15±\sqrt{15^{2}}}{2\times 30}$$

Take the square root of 15^{2}.

$$r=\frac{-15±15}{2\times 30}$$

Multiply 2 times 30.

$$r=\frac{-15±15}{60}$$

Now solve the equation r=\frac{-15±15}{60} when ± is plus. Add -15 to 15.

$$r=\frac{0}{60}$$

Divide 0 by 60.

$$r=0$$

Now solve the equation r=\frac{-15±15}{60} when ± is minus. Subtract 15 from -15.

$$r=-\frac{30}{60}$$

Reduce the fraction \frac{-30}{60} to lowest terms by extracting and canceling out 30.

$$r=-\frac{1}{2}$$

The equation is now solved.

$$r=0 r=-\frac{1}{2}$$

Steps for Completing the Square

Add 15r to both sides.

$$30r^{2}+15r=0$$

Divide both sides by 30.

$$\frac{30r^{2}+15r}{30}=\frac{0}{30}$$

Dividing by 30 undoes the multiplication by 30.

$$r^{2}+\frac{15}{30}r=\frac{0}{30}$$

Reduce the fraction \frac{15}{30} to lowest terms by extracting and canceling out 15.

$$r^{2}+\frac{1}{2}r=\frac{0}{30}$$

Divide 0 by 30.

$$r^{2}+\frac{1}{2}r=0$$

Divide \frac{1}{2}, the coefficient of the x term, by 2 to get \frac{1}{4}. Then add the square of \frac{1}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$r^{2}+\frac{1}{2}r+\left(\frac{1}{4}\right)^{2}=\left(\frac{1}{4}\right)^{2}$$

Square \frac{1}{4} by squaring both the numerator and the denominator of the fraction.

$$r^{2}+\frac{1}{2}r+\frac{1}{16}=\frac{1}{16}$$

Factor r^{2}+\frac{1}{2}r+\frac{1}{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}.

$$\left(r+\frac{1}{4}\right)^{2}=\frac{1}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(r+\frac{1}{4}\right)^{2}}=\sqrt{\frac{1}{16}}$$

Simplify.

$$r+\frac{1}{4}=\frac{1}{4} r+\frac{1}{4}=-\frac{1}{4}$$

Subtract \frac{1}{4} from both sides of the equation.

$$r=0 r=-\frac{1}{2}$$