Solve (x-60)(120-0.5x)=1600

$$( x - 60 ) ( 120 - 0.5 x ) = 1600$$

Solve for x

$$x=80$$

$$x=220$$

Steps Using the Quadratic Formula

Use the distributive property to multiply x-60 by 120-0.5x and combine like terms.

$$150x-0.5x^{2}-7200=1600$$

Subtract 1600 from both sides.

$$150x-0.5x^{2}-7200-1600=0$$

Subtract 1600 from -7200 to get -8800.

$$150x-0.5x^{2}-8800=0$$

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.

$$-0.5x^{2}+150x-8800=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -0.5 for a, 150 for b, and -8800 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-150±\sqrt{150^{2}-4\left(-0.5\right)\left(-8800\right)}}{2\left(-0.5\right)}$$

Square 150.

$$x=\frac{-150±\sqrt{22500-4\left(-0.5\right)\left(-8800\right)}}{2\left(-0.5\right)}$$

Multiply -4 times -0.5.

$$x=\frac{-150±\sqrt{22500+2\left(-8800\right)}}{2\left(-0.5\right)}$$

Multiply 2 times -8800.

$$x=\frac{-150±\sqrt{22500-17600}}{2\left(-0.5\right)}$$

Add 22500 to -17600.

$$x=\frac{-150±\sqrt{4900}}{2\left(-0.5\right)}$$

Take the square root of 4900.

$$x=\frac{-150±70}{2\left(-0.5\right)}$$

Multiply 2 times -0.5.

$$x=\frac{-150±70}{-1}$$

Now solve the equation x=\frac{-150±70}{-1} when ± is plus. Add -150 to 70.

$$x=-\frac{80}{-1}$$

Divide -80 by -1.

$$x=80$$

Now solve the equation x=\frac{-150±70}{-1} when ± is minus. Subtract 70 from -150.

$$x=-\frac{220}{-1}$$

Divide -220 by -1.

$$x=220$$

The equation is now solved.

$$x=80 x=220$$

Steps for Completing the Square

Use the distributive property to multiply x-60 by 120-0.5x and combine like terms.

$$150x-0.5x^{2}-7200=1600$$

Add 7200 to both sides.

$$150x-0.5x^{2}=1600+7200$$

Add 1600 and 7200 to get 8800.

$$150x-0.5x^{2}=8800$$

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.

$$-0.5x^{2}+150x=8800$$

Multiply both sides by -2.

$$\frac{-0.5x^{2}+150x}{-0.5}=\frac{8800}{-0.5}$$

Dividing by -0.5 undoes the multiplication by -0.5.

$$x^{2}+\frac{150}{-0.5}x=\frac{8800}{-0.5}$$

Divide 150 by -0.5 by multiplying 150 by the reciprocal of -0.5.

$$x^{2}-300x=\frac{8800}{-0.5}$$

Divide 8800 by -0.5 by multiplying 8800 by the reciprocal of -0.5.

$$x^{2}-300x=-17600$$

Divide -300, the coefficient of the x term, by 2 to get -150. Then add the square of -150 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-300x+\left(-150\right)^{2}=-17600+\left(-150\right)^{2}$$

Square -150.

$$x^{2}-300x+22500=-17600+22500$$

Add -17600 to 22500.

$$x^{2}-300x+22500=4900$$

Factor x^{2}-300x+22500. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-150\right)^{2}=4900$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-150\right)^{2}}=\sqrt{4900}$$

Simplify.

$$x-150=70 x-150=-70$$

Add 150 to both sides of the equation.

$$x=220 x=80$$