Solve x(16-0.5x)-120=0

$$x(16-0.5x)-120=0$$

Solve for x

$$x=12$$

$$x=20$$

Steps Using the Quadratic Formula

Use the distributive property to multiply x by 16-0.5x.

$$16x-0.5x^{2}-120=0$$

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.

$$-0.5x^{2}+16x-120=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -0.5 for a, 16 for b, and -120 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-16±\sqrt{16^{2}-4\left(-0.5\right)\left(-120\right)}}{2\left(-0.5\right)}$$

Square 16.

$$x=\frac{-16±\sqrt{256-4\left(-0.5\right)\left(-120\right)}}{2\left(-0.5\right)}$$

Multiply -4 times -0.5.

$$x=\frac{-16±\sqrt{256+2\left(-120\right)}}{2\left(-0.5\right)}$$

Multiply 2 times -120.

$$x=\frac{-16±\sqrt{256-240}}{2\left(-0.5\right)}$$

Add 256 to -240.

$$x=\frac{-16±\sqrt{16}}{2\left(-0.5\right)}$$

Take the square root of 16.

$$x=\frac{-16±4}{2\left(-0.5\right)}$$

Multiply 2 times -0.5.

$$x=\frac{-16±4}{-1}$$

Now solve the equation x=\frac{-16±4}{-1} when ± is plus. Add -16 to 4.

$$x=-\frac{12}{-1}$$

Divide -12 by -1.

$$x=12$$

Now solve the equation x=\frac{-16±4}{-1} when ± is minus. Subtract 4 from -16.

$$x=-\frac{20}{-1}$$

Divide -20 by -1.

$$x=20$$

The equation is now solved.

$$x=12 x=20$$

Steps for Completing the Square

Use the distributive property to multiply x by 16-0.5x.

$$16x-0.5x^{2}-120=0$$

Add 120 to both sides. Anything plus zero gives itself.

$$16x-0.5x^{2}=120$$

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.

$$-0.5x^{2}+16x=120$$

Multiply both sides by -2.

$$\frac{-0.5x^{2}+16x}{-0.5}=\frac{120}{-0.5}$$

Dividing by -0.5 undoes the multiplication by -0.5.

$$x^{2}+\frac{16}{-0.5}x=\frac{120}{-0.5}$$

Divide 16 by -0.5 by multiplying 16 by the reciprocal of -0.5.

$$x^{2}-32x=\frac{120}{-0.5}$$

Divide 120 by -0.5 by multiplying 120 by the reciprocal of -0.5.

$$x^{2}-32x=-240$$

Divide -32, the coefficient of the x term, by 2 to get -16. Then add the square of -16 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-32x+\left(-16\right)^{2}=-240+\left(-16\right)^{2}$$

Square -16.

$$x^{2}-32x+256=-240+256$$

Add -240 to 256.

$$x^{2}-32x+256=16$$

Factor x^{2}-32x+256. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

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

Take the square root of both sides of the equation.

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

Simplify.

$$x-16=4 x-16=-4$$

Add 16 to both sides of the equation.

$$x=20 x=12$$