Solve (100-2x)(50-2x)=3600

$$( 100 - 2 x ) ( 50 - 2 x ) = 3600$$

Solve for x

$$x=70$$

$$x=5$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 100-2x by 50-2x and combine like terms.

$$5000-300x+4x^{2}=3600$$

Subtract 3600 from both sides.

$$5000-300x+4x^{2}-3600=0$$

Subtract 3600 from 5000 to get 1400.

$$1400-300x+4x^{2}=0$$

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.

$$4x^{2}-300x+1400=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, -300 for b, and 1400 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-300\right)±\sqrt{\left(-300\right)^{2}-4\times 4\times 1400}}{2\times 4}$$

Square -300.

$$x=\frac{-\left(-300\right)±\sqrt{90000-4\times 4\times 1400}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-\left(-300\right)±\sqrt{90000-16\times 1400}}{2\times 4}$$

Multiply -16 times 1400.

$$x=\frac{-\left(-300\right)±\sqrt{90000-22400}}{2\times 4}$$

Add 90000 to -22400.

$$x=\frac{-\left(-300\right)±\sqrt{67600}}{2\times 4}$$

Take the square root of 67600.

$$x=\frac{-\left(-300\right)±260}{2\times 4}$$

The opposite of -300 is 300.

$$x=\frac{300±260}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{300±260}{8}$$

Now solve the equation x=\frac{300±260}{8} when ± is plus. Add 300 to 260.

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

Divide 560 by 8.

$$x=70$$

Now solve the equation x=\frac{300±260}{8} when ± is minus. Subtract 260 from 300.

$$x=\frac{40}{8}$$

Divide 40 by 8.

$$x=5$$

The equation is now solved.

$$x=70 x=5$$

Steps for Completing the Square

Use the distributive property to multiply 100-2x by 50-2x and combine like terms.

$$5000-300x+4x^{2}=3600$$

Subtract 5000 from both sides.

$$-300x+4x^{2}=3600-5000$$

Subtract 5000 from 3600 to get -1400.

$$-300x+4x^{2}=-1400$$

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.

$$4x^{2}-300x=-1400$$

Divide both sides by 4.

$$\frac{4x^{2}-300x}{4}=-\frac{1400}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$x^{2}+\left(-\frac{300}{4}\right)x=-\frac{1400}{4}$$

Divide -300 by 4.

$$x^{2}-75x=-\frac{1400}{4}$$

Divide -1400 by 4.

$$x^{2}-75x=-350$$

Divide -75, the coefficient of the x term, by 2 to get -\frac{75}{2}. Then add the square of -\frac{75}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-75x+\left(-\frac{75}{2}\right)^{2}=-350+\left(-\frac{75}{2}\right)^{2}$$

Square -\frac{75}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-75x+\frac{5625}{4}=-350+\frac{5625}{4}$$

Add -350 to \frac{5625}{4}.

$$x^{2}-75x+\frac{5625}{4}=\frac{4225}{4}$$

Factor x^{2}-75x+\frac{5625}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{75}{2}\right)^{2}=\frac{4225}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{75}{2}\right)^{2}}=\sqrt{\frac{4225}{4}}$$

Simplify.

$$x-\frac{75}{2}=\frac{65}{2} x-\frac{75}{2}=-\frac{65}{2}$$

Add \frac{75}{2} to both sides of the equation.

$$x=70 x=5$$