Solve 72y=-40y^2

$$72 y = - 40 y ^ { 2 }$$

Solve for y

$$y = -\frac{9}{5} = -1\frac{4}{5} = -1.8$$

$$y=0$$

Steps Using Factoring

Add 40y^{2} to both sides.

$$72y+40y^{2}=0$$

Factor out y.

$$y\left(72+40y\right)=0$$

To find equation solutions, solve y=0 and 72+40y=0.

$$y=0 y=-\frac{9}{5}$$

Steps Using the Quadratic Formula

Add 40y^{2} to both sides.

$$72y+40y^{2}=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$40y^{2}+72y=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 40 for a, 72 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-72±\sqrt{72^{2}}}{2\times 40}$$

Take the square root of 72^{2}.

$$y=\frac{-72±72}{2\times 40}$$

Multiply 2 times 40.

$$y=\frac{-72±72}{80}$$

Now solve the equation y=\frac{-72±72}{80} when ± is plus. Add -72 to 72.

$$y=\frac{0}{80}$$

Divide 0 by 80.

$$y=0$$

Now solve the equation y=\frac{-72±72}{80} when ± is minus. Subtract 72 from -72.

$$y=-\frac{144}{80}$$

Reduce the fraction \frac{-144}{80} to lowest terms by extracting and canceling out 16.

$$y=-\frac{9}{5}$$

The equation is now solved.

$$y=0 y=-\frac{9}{5}$$

Steps for Completing the Square

Add 40y^{2} to both sides.

$$72y+40y^{2}=0$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$40y^{2}+72y=0$$

Divide both sides by 40.

$$\frac{40y^{2}+72y}{40}=\frac{0}{40}$$

Dividing by 40 undoes the multiplication by 40.

$$y^{2}+\frac{72}{40}y=\frac{0}{40}$$

Reduce the fraction \frac{72}{40} to lowest terms by extracting and canceling out 8.

$$y^{2}+\frac{9}{5}y=\frac{0}{40}$$

Divide 0 by 40.

$$y^{2}+\frac{9}{5}y=0$$

Divide \frac{9}{5}, the coefficient of the x term, by 2 to get \frac{9}{10}. Then add the square of \frac{9}{10} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$y^{2}+\frac{9}{5}y+\left(\frac{9}{10}\right)^{2}=\left(\frac{9}{10}\right)^{2}$$

Square \frac{9}{10} by squaring both the numerator and the denominator of the fraction.

$$y^{2}+\frac{9}{5}y+\frac{81}{100}=\frac{81}{100}$$

Factor y^{2}+\frac{9}{5}y+\frac{81}{100}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(y+\frac{9}{10}\right)^{2}=\frac{81}{100}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y+\frac{9}{10}\right)^{2}}=\sqrt{\frac{81}{100}}$$

Simplify.

$$y+\frac{9}{10}=\frac{9}{10} y+\frac{9}{10}=-\frac{9}{10}$$

Subtract \frac{9}{10} from both sides of the equation.

$$y=0 y=-\frac{9}{5}$$