Solve (40-2x)(250-2x)=7200

$$( 40 - 2 x ) ( 250 - 2 x ) = 7200$$

Solve for x

$$x=140$$

$$x=5$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 40-2x by 250-2x and combine like terms.

$$10000-580x+4x^{2}=7200$$

Subtract 7200 from both sides.

$$10000-580x+4x^{2}-7200=0$$

Subtract 7200 from 10000 to get 2800.

$$2800-580x+4x^{2}=0$$

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.

$$4x^{2}-580x+2800=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, -580 for b, and 2800 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-580\right)±\sqrt{\left(-580\right)^{2}-4\times 4\times 2800}}{2\times 4}$$

Square -580.

$$x=\frac{-\left(-580\right)±\sqrt{336400-4\times 4\times 2800}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-\left(-580\right)±\sqrt{336400-16\times 2800}}{2\times 4}$$

Multiply -16 times 2800.

$$x=\frac{-\left(-580\right)±\sqrt{336400-44800}}{2\times 4}$$

Add 336400 to -44800.

$$x=\frac{-\left(-580\right)±\sqrt{291600}}{2\times 4}$$

Take the square root of 291600.

$$x=\frac{-\left(-580\right)±540}{2\times 4}$$

The opposite of -580 is 580.

$$x=\frac{580±540}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{580±540}{8}$$

Now solve the equation x=\frac{580±540}{8} when ± is plus. Add 580 to 540.

$$x=\frac{1120}{8}$$

Divide 1120 by 8.

$$x=140$$

Now solve the equation x=\frac{580±540}{8} when ± is minus. Subtract 540 from 580.

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

Divide 40 by 8.

$$x=5$$

The equation is now solved.

$$x=140 x=5$$

Steps for Completing the Square

Use the distributive property to multiply 40-2x by 250-2x and combine like terms.

$$10000-580x+4x^{2}=7200$$

Subtract 10000 from both sides.

$$-580x+4x^{2}=7200-10000$$

Subtract 10000 from 7200 to get -2800.

$$-580x+4x^{2}=-2800$$

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.

$$4x^{2}-580x=-2800$$

Divide both sides by 4.

$$\frac{4x^{2}-580x}{4}=-\frac{2800}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$x^{2}+\left(-\frac{580}{4}\right)x=-\frac{2800}{4}$$

Divide -580 by 4.

$$x^{2}-145x=-\frac{2800}{4}$$

Divide -2800 by 4.

$$x^{2}-145x=-700$$

Divide -145, the coefficient of the x term, by 2 to get -\frac{145}{2}. Then add the square of -\frac{145}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-145x+\left(-\frac{145}{2}\right)^{2}=-700+\left(-\frac{145}{2}\right)^{2}$$

Square -\frac{145}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-145x+\frac{21025}{4}=-700+\frac{21025}{4}$$

Add -700 to \frac{21025}{4}.

$$x^{2}-145x+\frac{21025}{4}=\frac{18225}{4}$$

Factor x^{2}-145x+\frac{21025}{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}.

$$\left(x-\frac{145}{2}\right)^{2}=\frac{18225}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{145}{2}\right)^{2}}=\sqrt{\frac{18225}{4}}$$

Simplify.

$$x-\frac{145}{2}=\frac{135}{2} x-\frac{145}{2}=-\frac{135}{2}$$

Add \frac{145}{2} to both sides of the equation.

$$x=140 x=5$$