Solve 145800=-2L^2+1080L

$$145800 = - 2 L ^ { 2 } + 1080 L$$

Solve for L

$$L=270$$

Steps Using the Quadratic Formula

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

$$-2L^{2}+1080L=145800$$

Subtract 145800 from both sides.

$$-2L^{2}+1080L-145800=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 1080 for b, and -145800 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$L=\frac{-1080±\sqrt{1080^{2}-4\left(-2\right)\left(-145800\right)}}{2\left(-2\right)}$$

Square 1080.

$$L=\frac{-1080±\sqrt{1166400-4\left(-2\right)\left(-145800\right)}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$L=\frac{-1080±\sqrt{1166400+8\left(-145800\right)}}{2\left(-2\right)}$$

Multiply 8 times -145800.

$$L=\frac{-1080±\sqrt{1166400-1166400}}{2\left(-2\right)}$$

Add 1166400 to -1166400.

$$L=\frac{-1080±\sqrt{0}}{2\left(-2\right)}$$

Take the square root of 0.

$$L=-\frac{1080}{2\left(-2\right)}$$

Multiply 2 times -2.

$$L=-\frac{1080}{-4}$$

Divide -1080 by -4.

$$L=270$$

Steps for Completing the Square

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

$$-2L^{2}+1080L=145800$$

Divide both sides by -2.

$$\frac{-2L^{2}+1080L}{-2}=\frac{145800}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$L^{2}+\frac{1080}{-2}L=\frac{145800}{-2}$$

Divide 1080 by -2.

$$L^{2}-540L=\frac{145800}{-2}$$

Divide 145800 by -2.

$$L^{2}-540L=-72900$$

Divide -540, the coefficient of the x term, by 2 to get -270. Then add the square of -270 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$L^{2}-540L+\left(-270\right)^{2}=-72900+\left(-270\right)^{2}$$

Square -270.

$$L^{2}-540L+72900=-72900+72900$$

Add -72900 to 72900.

$$L^{2}-540L+72900=0$$

Factor L^{2}-540L+72900. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(L-270\right)^{2}=0$$

Take the square root of both sides of the equation.

$$\sqrt{\left(L-270\right)^{2}}=\sqrt{0}$$

Simplify.

$$L-270=0 L-270=0$$

Add 270 to both sides of the equation.

$$L=270 L=270$$

The equation is now solved. Solutions are the same.

$$L=270$$