Solve -96=y^2-40y

$$- 96 = y ^ { 2 } - 40 y$$

Solve for y

$$y=4\sqrt{19}+20\approx 37.435595774$$

$$y=20-4\sqrt{19}\approx 2.564404226$$

Steps Using the Quadratic Formula

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

$$y^{2}-40y=-96$$

Add 96 to both sides.

$$y^{2}-40y+96=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -40 for b, and 96 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\left(-40\right)±\sqrt{\left(-40\right)^{2}-4\times 96}}{2}$$

Square -40.

$$y=\frac{-\left(-40\right)±\sqrt{1600-4\times 96}}{2}$$

Multiply -4 times 96.

$$y=\frac{-\left(-40\right)±\sqrt{1600-384}}{2}$$

Add 1600 to -384.

$$y=\frac{-\left(-40\right)±\sqrt{1216}}{2}$$

Take the square root of 1216.

$$y=\frac{-\left(-40\right)±8\sqrt{19}}{2}$$

The opposite of -40 is 40.

$$y=\frac{40±8\sqrt{19}}{2}$$

Now solve the equation y=\frac{40±8\sqrt{19}}{2} when ± is plus. Add 40 to 8\sqrt{19}.

$$y=\frac{8\sqrt{19}+40}{2}$$

Divide 40+8\sqrt{19} by 2.

$$y=4\sqrt{19}+20$$

Now solve the equation y=\frac{40±8\sqrt{19}}{2} when ± is minus. Subtract 8\sqrt{19} from 40.

$$y=\frac{40-8\sqrt{19}}{2}$$

Divide 40-8\sqrt{19} by 2.

$$y=20-4\sqrt{19}$$

The equation is now solved.

$$y=4\sqrt{19}+20 y=20-4\sqrt{19}$$

Steps for Completing the Square

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

$$y^{2}-40y=-96$$

Divide -40, the coefficient of the x term, by 2 to get -20. Then add the square of -20 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$y^{2}-40y+\left(-20\right)^{2}=-96+\left(-20\right)^{2}$$

Square -20.

$$y^{2}-40y+400=-96+400$$

Add -96 to 400.

$$y^{2}-40y+400=304$$

Factor y^{2}-40y+400. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(y-20\right)^{2}=304$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y-20\right)^{2}}=\sqrt{304}$$

Simplify.

$$y-20=4\sqrt{19} y-20=-4\sqrt{19}$$

Add 20 to both sides of the equation.

$$y=4\sqrt{19}+20 y=20-4\sqrt{19}$$