Solve (16-x)x=120

$$( 16 - x ) x = 120$$

Solve for x (complex solution)

$$x=-2\sqrt{14}i+8\approx 8-7.483314774i$$

$$x=8+2\sqrt{14}i\approx 8+7.483314774i$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 16-x by x.

$$16x-x^{2}=120$$

Subtract 120 from both sides.

$$16x-x^{2}-120=0$$

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.

$$-x^{2}+16x-120=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 16 for b, and -120 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-16±\sqrt{16^{2}-4\left(-1\right)\left(-120\right)}}{2\left(-1\right)}$$

Square 16.

$$x=\frac{-16±\sqrt{256-4\left(-1\right)\left(-120\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-16±\sqrt{256+4\left(-120\right)}}{2\left(-1\right)}$$

Multiply 4 times -120.

$$x=\frac{-16±\sqrt{256-480}}{2\left(-1\right)}$$

Add 256 to -480.

$$x=\frac{-16±\sqrt{-224}}{2\left(-1\right)}$$

Take the square root of -224.

$$x=\frac{-16±4\sqrt{14}i}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-16±4\sqrt{14}i}{-2}$$

Now solve the equation x=\frac{-16±4\sqrt{14}i}{-2} when ± is plus. Add -16 to 4i\sqrt{14}.

$$x=\frac{-16+4\sqrt{14}i}{-2}$$

Divide -16+4i\sqrt{14} by -2.

$$x=-2\sqrt{14}i+8$$

Now solve the equation x=\frac{-16±4\sqrt{14}i}{-2} when ± is minus. Subtract 4i\sqrt{14} from -16.

$$x=\frac{-4\sqrt{14}i-16}{-2}$$

Divide -16-4i\sqrt{14} by -2.

$$x=8+2\sqrt{14}i$$

The equation is now solved.

$$x=-2\sqrt{14}i+8 x=8+2\sqrt{14}i$$

Steps for Completing the Square

Use the distributive property to multiply 16-x by x.

$$16x-x^{2}=120$$

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.

$$-x^{2}+16x=120$$

Divide both sides by -1.

$$\frac{-x^{2}+16x}{-1}=\frac{120}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{16}{-1}x=\frac{120}{-1}$$

Divide 16 by -1.

$$x^{2}-16x=\frac{120}{-1}$$

Divide 120 by -1.

$$x^{2}-16x=-120$$

Divide -16, the coefficient of the x term, by 2 to get -8. Then add the square of -8 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-16x+\left(-8\right)^{2}=-120+\left(-8\right)^{2}$$

Square -8.

$$x^{2}-16x+64=-120+64$$

Add -120 to 64.

$$x^{2}-16x+64=-56$$

Factor x^{2}-16x+64. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-8\right)^{2}=-56$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-8\right)^{2}}=\sqrt{-56}$$

Simplify.

$$x-8=2\sqrt{14}i x-8=-2\sqrt{14}i$$

Add 8 to both sides of the equation.

$$x=8+2\sqrt{14}i x=-2\sqrt{14}i+8$$