Solve 1560=-x^2+186x

$$1560=- { x }^{ 2 } +186x$$

Solve for x

$$x=\sqrt{7089}+93\approx 177.196199439$$

$$x=93-\sqrt{7089}\approx 8.803800561$$

Steps Using the Quadratic Formula

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

$$-x^{2}+186x=1560$$

Subtract 1560 from both sides.

$$-x^{2}+186x-1560=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 186 for b, and -1560 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-186±\sqrt{186^{2}-4\left(-1\right)\left(-1560\right)}}{2\left(-1\right)}$$

Square 186.

$$x=\frac{-186±\sqrt{34596-4\left(-1\right)\left(-1560\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-186±\sqrt{34596+4\left(-1560\right)}}{2\left(-1\right)}$$

Multiply 4 times -1560.

$$x=\frac{-186±\sqrt{34596-6240}}{2\left(-1\right)}$$

Add 34596 to -6240.

$$x=\frac{-186±\sqrt{28356}}{2\left(-1\right)}$$

Take the square root of 28356.

$$x=\frac{-186±2\sqrt{7089}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-186±2\sqrt{7089}}{-2}$$

Now solve the equation x=\frac{-186±2\sqrt{7089}}{-2} when ± is plus. Add -186 to 2\sqrt{7089}.

$$x=\frac{2\sqrt{7089}-186}{-2}$$

Divide -186+2\sqrt{7089} by -2.

$$x=93-\sqrt{7089}$$

Now solve the equation x=\frac{-186±2\sqrt{7089}}{-2} when ± is minus. Subtract 2\sqrt{7089} from -186.

$$x=\frac{-2\sqrt{7089}-186}{-2}$$

Divide -186-2\sqrt{7089} by -2.

$$x=\sqrt{7089}+93$$

The equation is now solved.

$$x=93-\sqrt{7089} x=\sqrt{7089}+93$$

Steps for Completing the Square

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

$$-x^{2}+186x=1560$$

Divide both sides by -1.

$$\frac{-x^{2}+186x}{-1}=\frac{1560}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{186}{-1}x=\frac{1560}{-1}$$

Divide 186 by -1.

$$x^{2}-186x=\frac{1560}{-1}$$

Divide 1560 by -1.

$$x^{2}-186x=-1560$$

Divide -186, the coefficient of the x term, by 2 to get -93. Then add the square of -93 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-186x+\left(-93\right)^{2}=-1560+\left(-93\right)^{2}$$

Square -93.

$$x^{2}-186x+8649=-1560+8649$$

Add -1560 to 8649.

$$x^{2}-186x+8649=7089$$

Factor x^{2}-186x+8649. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-93\right)^{2}=7089$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-93\right)^{2}}=\sqrt{7089}$$

Simplify.

$$x-93=\sqrt{7089} x-93=-\sqrt{7089}$$

Add 93 to both sides of the equation.

$$x=\sqrt{7089}+93 x=93-\sqrt{7089}$$