Solve (x-120)(240-x)=3200

$$( x - 120 ) ( 240 - x ) = 3200$$

Solve for x

$$x=160$$

$$x=200$$

Steps Using the Quadratic Formula

Use the distributive property to multiply x-120 by 240-x and combine like terms.

$$360x-x^{2}-28800=3200$$

Subtract 3200 from both sides.

$$360x-x^{2}-28800-3200=0$$

Subtract 3200 from -28800 to get -32000.

$$360x-x^{2}-32000=0$$

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.

$$-x^{2}+360x-32000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 360 for b, and -32000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-360±\sqrt{360^{2}-4\left(-1\right)\left(-32000\right)}}{2\left(-1\right)}$$

Square 360.

$$x=\frac{-360±\sqrt{129600-4\left(-1\right)\left(-32000\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-360±\sqrt{129600+4\left(-32000\right)}}{2\left(-1\right)}$$

Multiply 4 times -32000.

$$x=\frac{-360±\sqrt{129600-128000}}{2\left(-1\right)}$$

Add 129600 to -128000.

$$x=\frac{-360±\sqrt{1600}}{2\left(-1\right)}$$

Take the square root of 1600.

$$x=\frac{-360±40}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-360±40}{-2}$$

Now solve the equation x=\frac{-360±40}{-2} when ± is plus. Add -360 to 40.

$$x=-\frac{320}{-2}$$

Divide -320 by -2.

$$x=160$$

Now solve the equation x=\frac{-360±40}{-2} when ± is minus. Subtract 40 from -360.

$$x=-\frac{400}{-2}$$

Divide -400 by -2.

$$x=200$$

The equation is now solved.

$$x=160 x=200$$

Steps for Completing the Square

Use the distributive property to multiply x-120 by 240-x and combine like terms.

$$360x-x^{2}-28800=3200$$

Add 28800 to both sides.

$$360x-x^{2}=3200+28800$$

Add 3200 and 28800 to get 32000.

$$360x-x^{2}=32000$$

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.

$$-x^{2}+360x=32000$$

Divide both sides by -1.

$$\frac{-x^{2}+360x}{-1}=\frac{32000}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{360}{-1}x=\frac{32000}{-1}$$

Divide 360 by -1.

$$x^{2}-360x=\frac{32000}{-1}$$

Divide 32000 by -1.

$$x^{2}-360x=-32000$$

Divide -360, the coefficient of the x term, by 2 to get -180. Then add the square of -180 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-360x+\left(-180\right)^{2}=-32000+\left(-180\right)^{2}$$

Square -180.

$$x^{2}-360x+32400=-32000+32400$$

Add -32000 to 32400.

$$x^{2}-360x+32400=400$$

Factor x^{2}-360x+32400. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-180\right)^{2}=400$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-180\right)^{2}}=\sqrt{400}$$

Simplify.

$$x-180=20 x-180=-20$$

Add 180 to both sides of the equation.

$$x=200 x=160$$