Solve 160=-x^2+92x-1920

$$160 = - { x }^{ 2 } +92x-1920$$

Solve for x

$$x=40$$

$$x=52$$

Steps Using the Quadratic Formula

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

$$-x^{2}+92x-1920=160$$

Subtract 160 from both sides.

$$-x^{2}+92x-1920-160=0$$

Subtract 160 from -1920 to get -2080.

$$-x^{2}+92x-2080=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 92 for b, and -2080 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-92±\sqrt{92^{2}-4\left(-1\right)\left(-2080\right)}}{2\left(-1\right)}$$

Square 92.

$$x=\frac{-92±\sqrt{8464-4\left(-1\right)\left(-2080\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-92±\sqrt{8464+4\left(-2080\right)}}{2\left(-1\right)}$$

Multiply 4 times -2080.

$$x=\frac{-92±\sqrt{8464-8320}}{2\left(-1\right)}$$

Add 8464 to -8320.

$$x=\frac{-92±\sqrt{144}}{2\left(-1\right)}$$

Take the square root of 144.

$$x=\frac{-92±12}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-92±12}{-2}$$

Now solve the equation x=\frac{-92±12}{-2} when ± is plus. Add -92 to 12.

$$x=-\frac{80}{-2}$$

Divide -80 by -2.

$$x=40$$

Now solve the equation x=\frac{-92±12}{-2} when ± is minus. Subtract 12 from -92.

$$x=-\frac{104}{-2}$$

Divide -104 by -2.

$$x=52$$

The equation is now solved.

$$x=40 x=52$$

Steps for Completing the Square

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

$$-x^{2}+92x-1920=160$$

Add 1920 to both sides.

$$-x^{2}+92x=160+1920$$

Add 160 and 1920 to get 2080.

$$-x^{2}+92x=2080$$

Divide both sides by -1.

$$\frac{-x^{2}+92x}{-1}=\frac{2080}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{92}{-1}x=\frac{2080}{-1}$$

Divide 92 by -1.

$$x^{2}-92x=\frac{2080}{-1}$$

Divide 2080 by -1.

$$x^{2}-92x=-2080$$

Divide -92, the coefficient of the x term, by 2 to get -46. Then add the square of -46 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-92x+\left(-46\right)^{2}=-2080+\left(-46\right)^{2}$$

Square -46.

$$x^{2}-92x+2116=-2080+2116$$

Add -2080 to 2116.

$$x^{2}-92x+2116=36$$

Factor x^{2}-92x+2116. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-46\right)^{2}=36$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-46\right)^{2}}=\sqrt{36}$$

Simplify.

$$x-46=6 x-46=-6$$

Add 46 to both sides of the equation.

$$x=52 x=40$$