Solve (x-120)(200-x)=1500

$$(x-120)(200-x)=1500$$

Solve for x

$$x=150$$

$$x=170$$

Steps Using the Quadratic Formula

Use the distributive property to multiply x-120 by 200-x and combine like terms.

$$320x-x^{2}-24000=1500$$

Subtract 1500 from both sides.

$$320x-x^{2}-24000-1500=0$$

Subtract 1500 from -24000 to get -25500.

$$320x-x^{2}-25500=0$$

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.

$$-x^{2}+320x-25500=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 320 for b, and -25500 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-320±\sqrt{320^{2}-4\left(-1\right)\left(-25500\right)}}{2\left(-1\right)}$$

Square 320.

$$x=\frac{-320±\sqrt{102400-4\left(-1\right)\left(-25500\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-320±\sqrt{102400+4\left(-25500\right)}}{2\left(-1\right)}$$

Multiply 4 times -25500.

$$x=\frac{-320±\sqrt{102400-102000}}{2\left(-1\right)}$$

Add 102400 to -102000.

$$x=\frac{-320±\sqrt{400}}{2\left(-1\right)}$$

Take the square root of 400.

$$x=\frac{-320±20}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-320±20}{-2}$$

Now solve the equation x=\frac{-320±20}{-2} when ± is plus. Add -320 to 20.

$$x=-\frac{300}{-2}$$

Divide -300 by -2.

$$x=150$$

Now solve the equation x=\frac{-320±20}{-2} when ± is minus. Subtract 20 from -320.

$$x=-\frac{340}{-2}$$

Divide -340 by -2.

$$x=170$$

The equation is now solved.

$$x=150 x=170$$

Steps for Completing the Square

Use the distributive property to multiply x-120 by 200-x and combine like terms.

$$320x-x^{2}-24000=1500$$

Add 24000 to both sides.

$$320x-x^{2}=1500+24000$$

Add 1500 and 24000 to get 25500.

$$320x-x^{2}=25500$$

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.

$$-x^{2}+320x=25500$$

Divide both sides by -1.

$$\frac{-x^{2}+320x}{-1}=\frac{25500}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{320}{-1}x=\frac{25500}{-1}$$

Divide 320 by -1.

$$x^{2}-320x=\frac{25500}{-1}$$

Divide 25500 by -1.

$$x^{2}-320x=-25500$$

Divide -320, the coefficient of the x term, by 2 to get -160. Then add the square of -160 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-320x+\left(-160\right)^{2}=-25500+\left(-160\right)^{2}$$

Square -160.

$$x^{2}-320x+25600=-25500+25600$$

Add -25500 to 25600.

$$x^{2}-320x+25600=100$$

Factor x^{2}-320x+25600. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-160\right)^{2}=100$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-160\right)^{2}}=\sqrt{100}$$

Simplify.

$$x-160=10 x-160=-10$$

Add 160 to both sides of the equation.

$$x=170 x=150$$