Solve 120-4x=8+2x-x^2

$$120-4x=8+2x- { x }^{ 2 }$$

Solve for x (complex solution)

$$x=3+\sqrt{103}i\approx 3+10.148891565i$$

$$x=-\sqrt{103}i+3\approx 3-10.148891565i$$

Steps Using the Quadratic Formula

Subtract 8 from both sides.

$$120-4x-8=2x-x^{2}$$

Subtract 8 from 120 to get 112.

$$112-4x=2x-x^{2}$$

Subtract 2x from both sides.

$$112-4x-2x=-x^{2}$$

Combine -4x and -2x to get -6x.

$$112-6x=-x^{2}$$

Add x^{2} to both sides.

$$112-6x+x^{2}=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$x^{2}-6x+112=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -6 for b, and 112 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\times 112}}{2}$$

Square -6.

$$x=\frac{-\left(-6\right)±\sqrt{36-4\times 112}}{2}$$

Multiply -4 times 112.

$$x=\frac{-\left(-6\right)±\sqrt{36-448}}{2}$$

Add 36 to -448.

$$x=\frac{-\left(-6\right)±\sqrt{-412}}{2}$$

Take the square root of -412.

$$x=\frac{-\left(-6\right)±2\sqrt{103}i}{2}$$

The opposite of -6 is 6.

$$x=\frac{6±2\sqrt{103}i}{2}$$

Now solve the equation x=\frac{6±2\sqrt{103}i}{2} when ± is plus. Add 6 to 2i\sqrt{103}.

$$x=\frac{6+2\sqrt{103}i}{2}$$

Divide 6+2i\sqrt{103} by 2.

$$x=3+\sqrt{103}i$$

Now solve the equation x=\frac{6±2\sqrt{103}i}{2} when ± is minus. Subtract 2i\sqrt{103} from 6.

$$x=\frac{-2\sqrt{103}i+6}{2}$$

Divide 6-2i\sqrt{103} by 2.

$$x=-\sqrt{103}i+3$$

The equation is now solved.

$$x=3+\sqrt{103}i x=-\sqrt{103}i+3$$

Steps for Completing the Square

Subtract 2x from both sides.

$$120-4x-2x=8-x^{2}$$

Combine -4x and -2x to get -6x.

$$120-6x=8-x^{2}$$

Add x^{2} to both sides.

$$120-6x+x^{2}=8$$

Subtract 120 from both sides.

$$-6x+x^{2}=8-120$$

Subtract 120 from 8 to get -112.

$$-6x+x^{2}=-112$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$x^{2}-6x=-112$$

Divide -6, the coefficient of the x term, by 2 to get -3. Then add the square of -3 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-6x+\left(-3\right)^{2}=-112+\left(-3\right)^{2}$$

Square -3.

$$x^{2}-6x+9=-112+9$$

Add -112 to 9.

$$x^{2}-6x+9=-103$$

Factor x^{2}-6x+9. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-3\right)^{2}=-103$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-3\right)^{2}}=\sqrt{-103}$$

Simplify.

$$x-3=\sqrt{103}i x-3=-\sqrt{103}i$$

Add 3 to both sides of the equation.

$$x=3+\sqrt{103}i x=-\sqrt{103}i+3$$