Solve 3y^2=-21y

$$3 y ^ { 2 } = - 21 y$$

Solve for y

$$y=-7$$

$$y=0$$

Steps Using Factoring

Add 21y to both sides.

$$3y^{2}+21y=0$$

Factor out y.

$$y\left(3y+21\right)=0$$

To find equation solutions, solve y=0 and 3y+21=0.

$$y=0 y=-7$$

Steps Using the Quadratic Formula

Add 21y to both sides.

$$3y^{2}+21y=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, 21 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-21±\sqrt{21^{2}}}{2\times 3}$$

Take the square root of 21^{2}.

$$y=\frac{-21±21}{2\times 3}$$

Multiply 2 times 3.

$$y=\frac{-21±21}{6}$$

Now solve the equation y=\frac{-21±21}{6} when ± is plus. Add -21 to 21.

$$y=\frac{0}{6}$$

Divide 0 by 6.

$$y=0$$

Now solve the equation y=\frac{-21±21}{6} when ± is minus. Subtract 21 from -21.

$$y=-\frac{42}{6}$$

Divide -42 by 6.

$$y=-7$$

The equation is now solved.

$$y=0 y=-7$$

Steps for Completing the Square

Add 21y to both sides.

$$3y^{2}+21y=0$$

Divide both sides by 3.

$$\frac{3y^{2}+21y}{3}=\frac{0}{3}$$

Dividing by 3 undoes the multiplication by 3.

$$y^{2}+\frac{21}{3}y=\frac{0}{3}$$

Divide 21 by 3.

$$y^{2}+7y=\frac{0}{3}$$

Divide 0 by 3.

$$y^{2}+7y=0$$

Divide 7, the coefficient of the x term, by 2 to get \frac{7}{2}. Then add the square of \frac{7}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$y^{2}+7y+\left(\frac{7}{2}\right)^{2}=\left(\frac{7}{2}\right)^{2}$$

Square \frac{7}{2} by squaring both the numerator and the denominator of the fraction.

$$y^{2}+7y+\frac{49}{4}=\frac{49}{4}$$

Factor y^{2}+7y+\frac{49}{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}.

$$\left(y+\frac{7}{2}\right)^{2}=\frac{49}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y+\frac{7}{2}\right)^{2}}=\sqrt{\frac{49}{4}}$$

Simplify.

$$y+\frac{7}{2}=\frac{7}{2} y+\frac{7}{2}=-\frac{7}{2}$$

Subtract \frac{7}{2} from both sides of the equation.

$$y=0 y=-7$$