Solve x*(24-x/2)=80

$$x \cdot ( \frac { 24 - x } { 2 } ) = 80$$

Solve for x (complex solution)

$$x=12-4i$$

$$x=12+4i$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 2.

$$x\left(24-x\right)=160$$

Use the distributive property to multiply x by 24-x.

$$24x-x^{2}=160$$

Subtract 160 from both sides.

$$24x-x^{2}-160=0$$

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.

$$-x^{2}+24x-160=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 24 for b, and -160 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-24±\sqrt{24^{2}-4\left(-1\right)\left(-160\right)}}{2\left(-1\right)}$$

Square 24.

$$x=\frac{-24±\sqrt{576-4\left(-1\right)\left(-160\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-24±\sqrt{576+4\left(-160\right)}}{2\left(-1\right)}$$

Multiply 4 times -160.

$$x=\frac{-24±\sqrt{576-640}}{2\left(-1\right)}$$

Add 576 to -640.

$$x=\frac{-24±\sqrt{-64}}{2\left(-1\right)}$$

Take the square root of -64.

$$x=\frac{-24±8i}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-24±8i}{-2}$$

Now solve the equation x=\frac{-24±8i}{-2} when ± is plus. Add -24 to 8i.

$$x=\frac{-24+8i}{-2}$$

Divide -24+8i by -2.

$$x=12-4i$$

Now solve the equation x=\frac{-24±8i}{-2} when ± is minus. Subtract 8i from -24.

$$x=\frac{-24-8i}{-2}$$

Divide -24-8i by -2.

$$x=12+4i$$

The equation is now solved.

$$x=12-4i x=12+4i$$

Steps for Completing the Square

Multiply both sides of the equation by 2.

$$x\left(24-x\right)=160$$

Use the distributive property to multiply x by 24-x.

$$24x-x^{2}=160$$

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.

$$-x^{2}+24x=160$$

Divide both sides by -1.

$$\frac{-x^{2}+24x}{-1}=\frac{160}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{24}{-1}x=\frac{160}{-1}$$

Divide 24 by -1.

$$x^{2}-24x=\frac{160}{-1}$$

Divide 160 by -1.

$$x^{2}-24x=-160$$

Divide -24, the coefficient of the x term, by 2 to get -12. Then add the square of -12 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-24x+\left(-12\right)^{2}=-160+\left(-12\right)^{2}$$

Square -12.

$$x^{2}-24x+144=-160+144$$

Add -160 to 144.

$$x^{2}-24x+144=-16$$

Factor x^{2}-24x+144. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-12\right)^{2}=-16$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-12\right)^{2}}=\sqrt{-16}$$

Simplify.

$$x-12=4i x-12=-4i$$

Add 12 to both sides of the equation.

$$x=12+4i x=12-4i$$