Solve -24y^2=9y

$$- 24 y ^ { 2 } = 9 y$$

Solve for y

$$y=-\frac{3}{8}=-0.375$$

$$y=0$$

Steps Using Factoring

Subtract 9y from both sides.

$$-24y^{2}-9y=0$$

Factor out y.

$$y\left(-24y-9\right)=0$$

To find equation solutions, solve y=0 and -24y-9=0.

$$y=0 y=-\frac{3}{8}$$

Steps Using the Quadratic Formula

Subtract 9y from both sides.

$$-24y^{2}-9y=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -24 for a, -9 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\left(-9\right)±\sqrt{\left(-9\right)^{2}}}{2\left(-24\right)}$$

Take the square root of \left(-9\right)^{2}.

$$y=\frac{-\left(-9\right)±9}{2\left(-24\right)}$$

The opposite of -9 is 9.

$$y=\frac{9±9}{2\left(-24\right)}$$

Multiply 2 times -24.

$$y=\frac{9±9}{-48}$$

Now solve the equation y=\frac{9±9}{-48} when ± is plus. Add 9 to 9.

$$y=\frac{18}{-48}$$

Reduce the fraction \frac{18}{-48} to lowest terms by extracting and canceling out 6.

$$y=-\frac{3}{8}$$

Now solve the equation y=\frac{9±9}{-48} when ± is minus. Subtract 9 from 9.

$$y=\frac{0}{-48}$$

Divide 0 by -48.

$$y=0$$

The equation is now solved.

$$y=-\frac{3}{8} y=0$$

Steps for Completing the Square

Subtract 9y from both sides.

$$-24y^{2}-9y=0$$

Divide both sides by -24.

$$\frac{-24y^{2}-9y}{-24}=\frac{0}{-24}$$

Dividing by -24 undoes the multiplication by -24.

$$y^{2}+\left(-\frac{9}{-24}\right)y=\frac{0}{-24}$$

Reduce the fraction \frac{-9}{-24} to lowest terms by extracting and canceling out 3.

$$y^{2}+\frac{3}{8}y=\frac{0}{-24}$$

Divide 0 by -24.

$$y^{2}+\frac{3}{8}y=0$$

Divide \frac{3}{8}, the coefficient of the x term, by 2 to get \frac{3}{16}. Then add the square of \frac{3}{16} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$y^{2}+\frac{3}{8}y+\left(\frac{3}{16}\right)^{2}=\left(\frac{3}{16}\right)^{2}$$

Square \frac{3}{16} by squaring both the numerator and the denominator of the fraction.

$$y^{2}+\frac{3}{8}y+\frac{9}{256}=\frac{9}{256}$$

Factor y^{2}+\frac{3}{8}y+\frac{9}{256}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(y+\frac{3}{16}\right)^{2}=\frac{9}{256}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y+\frac{3}{16}\right)^{2}}=\sqrt{\frac{9}{256}}$$

Simplify.

$$y+\frac{3}{16}=\frac{3}{16} y+\frac{3}{16}=-\frac{3}{16}$$

Subtract \frac{3}{16} from both sides of the equation.

$$y=0 y=-\frac{3}{8}$$