Solve y^2-24y+144y=-100

$${ y }^{ 2 } -24y+144y = -100$$

Solve for y

$$y=10\sqrt{35}-60\approx -0.839202169$$

$$y=-10\sqrt{35}-60\approx -119.160797831$$

Steps Using the Quadratic Formula

Combine -24y and 144y to get 120y.

$$y^{2}+120y=-100$$

Add 100 to both sides.

$$y^{2}+120y+100=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 120 for b, and 100 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-120±\sqrt{120^{2}-4\times 100}}{2}$$

Square 120.

$$y=\frac{-120±\sqrt{14400-4\times 100}}{2}$$

Multiply -4 times 100.

$$y=\frac{-120±\sqrt{14400-400}}{2}$$

Add 14400 to -400.

$$y=\frac{-120±\sqrt{14000}}{2}$$

Take the square root of 14000.

$$y=\frac{-120±20\sqrt{35}}{2}$$

Now solve the equation y=\frac{-120±20\sqrt{35}}{2} when ± is plus. Add -120 to 20\sqrt{35}.

$$y=\frac{20\sqrt{35}-120}{2}$$

Divide -120+20\sqrt{35} by 2.

$$y=10\sqrt{35}-60$$

Now solve the equation y=\frac{-120±20\sqrt{35}}{2} when ± is minus. Subtract 20\sqrt{35} from -120.

$$y=\frac{-20\sqrt{35}-120}{2}$$

Divide -120-20\sqrt{35} by 2.

$$y=-10\sqrt{35}-60$$

The equation is now solved.

$$y=10\sqrt{35}-60 y=-10\sqrt{35}-60$$

Steps for Completing the Square

Combine -24y and 144y to get 120y.

$$y^{2}+120y=-100$$

Divide 120, the coefficient of the x term, by 2 to get 60. Then add the square of 60 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$y^{2}+120y+60^{2}=-100+60^{2}$$

Square 60.

$$y^{2}+120y+3600=-100+3600$$

Add -100 to 3600.

$$y^{2}+120y+3600=3500$$

Factor y^{2}+120y+3600. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(y+60\right)^{2}=3500$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y+60\right)^{2}}=\sqrt{3500}$$

Simplify.

$$y+60=10\sqrt{35} y+60=-10\sqrt{35}$$

Subtract 60 from both sides of the equation.

$$y=10\sqrt{35}-60 y=-10\sqrt{35}-60$$