Solve -60=-2y^2+21y

$$- 60 = - 2 y ^ { 2 } + 21 y$$

Solve for y

$$y = \frac{\sqrt{921} + 21}{4} \approx 12.836995453$$

$$y=\frac{21-\sqrt{921}}{4}\approx -2.336995453$$

Steps Using the Quadratic Formula

Swap sides so that all variable terms are on the left hand side.

$$-2y^{2}+21y=-60$$

Add 60 to both sides.

$$-2y^{2}+21y+60=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 21 for b, and 60 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-21±\sqrt{21^{2}-4\left(-2\right)\times 60}}{2\left(-2\right)}$$

Square 21.

$$y=\frac{-21±\sqrt{441-4\left(-2\right)\times 60}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$y=\frac{-21±\sqrt{441+8\times 60}}{2\left(-2\right)}$$

Multiply 8 times 60.

$$y=\frac{-21±\sqrt{441+480}}{2\left(-2\right)}$$

Add 441 to 480.

$$y=\frac{-21±\sqrt{921}}{2\left(-2\right)}$$

Multiply 2 times -2.

$$y=\frac{-21±\sqrt{921}}{-4}$$

Now solve the equation y=\frac{-21±\sqrt{921}}{-4} when ± is plus. Add -21 to \sqrt{921}.

$$y=\frac{\sqrt{921}-21}{-4}$$

Divide -21+\sqrt{921} by -4.

$$y=\frac{21-\sqrt{921}}{4}$$

Now solve the equation y=\frac{-21±\sqrt{921}}{-4} when ± is minus. Subtract \sqrt{921} from -21.

$$y=\frac{-\sqrt{921}-21}{-4}$$

Divide -21-\sqrt{921} by -4.

$$y=\frac{\sqrt{921}+21}{4}$$

The equation is now solved.

$$y=\frac{21-\sqrt{921}}{4} y=\frac{\sqrt{921}+21}{4}$$

Steps for Completing the Square

Swap sides so that all variable terms are on the left hand side.

$$-2y^{2}+21y=-60$$

Divide both sides by -2.

$$\frac{-2y^{2}+21y}{-2}=-\frac{60}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$y^{2}+\frac{21}{-2}y=-\frac{60}{-2}$$

Divide 21 by -2.

$$y^{2}-\frac{21}{2}y=-\frac{60}{-2}$$

Divide -60 by -2.

$$y^{2}-\frac{21}{2}y=30$$

Divide -\frac{21}{2}, the coefficient of the x term, by 2 to get -\frac{21}{4}. Then add the square of -\frac{21}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$y^{2}-\frac{21}{2}y+\left(-\frac{21}{4}\right)^{2}=30+\left(-\frac{21}{4}\right)^{2}$$

Square -\frac{21}{4} by squaring both the numerator and the denominator of the fraction.

$$y^{2}-\frac{21}{2}y+\frac{441}{16}=30+\frac{441}{16}$$

Add 30 to \frac{441}{16}.

$$y^{2}-\frac{21}{2}y+\frac{441}{16}=\frac{921}{16}$$

Factor y^{2}-\frac{21}{2}y+\frac{441}{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}.

$$\left(y-\frac{21}{4}\right)^{2}=\frac{921}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y-\frac{21}{4}\right)^{2}}=\sqrt{\frac{921}{16}}$$

Simplify.

$$y-\frac{21}{4}=\frac{\sqrt{921}}{4} y-\frac{21}{4}=-\frac{\sqrt{921}}{4}$$

Add \frac{21}{4} to both sides of the equation.

$$y=\frac{\sqrt{921}+21}{4} y=\frac{21-\sqrt{921}}{4}$$