Solve (120-2x)(100-2x)=800

$$(120-2x)(100-2x)=800$$

Solve for x

$$x=70$$

$$x=40$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 120-2x by 100-2x and combine like terms.

$$12000-440x+4x^{2}=800$$

Subtract 800 from both sides.

$$12000-440x+4x^{2}-800=0$$

Subtract 800 from 12000 to get 11200.

$$11200-440x+4x^{2}=0$$

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.

$$4x^{2}-440x+11200=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, -440 for b, and 11200 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-440\right)±\sqrt{\left(-440\right)^{2}-4\times 4\times 11200}}{2\times 4}$$

Square -440.

$$x=\frac{-\left(-440\right)±\sqrt{193600-4\times 4\times 11200}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-\left(-440\right)±\sqrt{193600-16\times 11200}}{2\times 4}$$

Multiply -16 times 11200.

$$x=\frac{-\left(-440\right)±\sqrt{193600-179200}}{2\times 4}$$

Add 193600 to -179200.

$$x=\frac{-\left(-440\right)±\sqrt{14400}}{2\times 4}$$

Take the square root of 14400.

$$x=\frac{-\left(-440\right)±120}{2\times 4}$$

The opposite of -440 is 440.

$$x=\frac{440±120}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{440±120}{8}$$

Now solve the equation x=\frac{440±120}{8} when ± is plus. Add 440 to 120.

$$x=\frac{560}{8}$$

Divide 560 by 8.

$$x=70$$

Now solve the equation x=\frac{440±120}{8} when ± is minus. Subtract 120 from 440.

$$x=\frac{320}{8}$$

Divide 320 by 8.

$$x=40$$

The equation is now solved.

$$x=70 x=40$$

Steps for Completing the Square

Use the distributive property to multiply 120-2x by 100-2x and combine like terms.

$$12000-440x+4x^{2}=800$$

Subtract 12000 from both sides.

$$-440x+4x^{2}=800-12000$$

Subtract 12000 from 800 to get -11200.

$$-440x+4x^{2}=-11200$$

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.

$$4x^{2}-440x=-11200$$

Divide both sides by 4.

$$\frac{4x^{2}-440x}{4}=-\frac{11200}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$x^{2}+\left(-\frac{440}{4}\right)x=-\frac{11200}{4}$$

Divide -440 by 4.

$$x^{2}-110x=-\frac{11200}{4}$$

Divide -11200 by 4.

$$x^{2}-110x=-2800$$

Divide -110, the coefficient of the x term, by 2 to get -55. Then add the square of -55 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-110x+\left(-55\right)^{2}=-2800+\left(-55\right)^{2}$$

Square -55.

$$x^{2}-110x+3025=-2800+3025$$

Add -2800 to 3025.

$$x^{2}-110x+3025=225$$

Factor x^{2}-110x+3025. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-55\right)^{2}=225$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-55\right)^{2}}=\sqrt{225}$$

Simplify.

$$x-55=15 x-55=-15$$

Add 55 to both sides of the equation.

$$x=70 x=40$$