Solve 500=-x^2+8x

$$500 = - x ^ { 2 } + 8 x$$

Solve for x (complex solution)

$$x=4+22i$$

$$x=4-22i$$

Steps Using the Quadratic Formula

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

$$-x^{2}+8x=500$$

Subtract 500 from both sides.

$$-x^{2}+8x-500=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 8 for b, and -500 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-8±\sqrt{8^{2}-4\left(-1\right)\left(-500\right)}}{2\left(-1\right)}$$

Square 8.

$$x=\frac{-8±\sqrt{64-4\left(-1\right)\left(-500\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-8±\sqrt{64+4\left(-500\right)}}{2\left(-1\right)}$$

Multiply 4 times -500.

$$x=\frac{-8±\sqrt{64-2000}}{2\left(-1\right)}$$

Add 64 to -2000.

$$x=\frac{-8±\sqrt{-1936}}{2\left(-1\right)}$$

Take the square root of -1936.

$$x=\frac{-8±44i}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-8±44i}{-2}$$

Now solve the equation x=\frac{-8±44i}{-2} when ± is plus. Add -8 to 44i.

$$x=\frac{-8+44i}{-2}$$

Divide -8+44i by -2.

$$x=4-22i$$

Now solve the equation x=\frac{-8±44i}{-2} when ± is minus. Subtract 44i from -8.

$$x=\frac{-8-44i}{-2}$$

Divide -8-44i by -2.

$$x=4+22i$$

The equation is now solved.

$$x=4-22i x=4+22i$$

Steps for Completing the Square

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

$$-x^{2}+8x=500$$

Divide both sides by -1.

$$\frac{-x^{2}+8x}{-1}=\frac{500}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{8}{-1}x=\frac{500}{-1}$$

Divide 8 by -1.

$$x^{2}-8x=\frac{500}{-1}$$

Divide 500 by -1.

$$x^{2}-8x=-500$$

Divide -8, the coefficient of the x term, by 2 to get -4. Then add the square of -4 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-8x+\left(-4\right)^{2}=-500+\left(-4\right)^{2}$$

Square -4.

$$x^{2}-8x+16=-500+16$$

Add -500 to 16.

$$x^{2}-8x+16=-484$$

Factor x^{2}-8x+16. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-4\right)^{2}=-484$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-4\right)^{2}}=\sqrt{-484}$$

Simplify.

$$x-4=22i x-4=-22i$$

Add 4 to both sides of the equation.

$$x=4+22i x=4-22i$$