Solve (280-2x)(160-2x)=16000

$$(280-2x)(160-2x)=16000$$

Solve for x

$$x=180$$

$$x=40$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 280-2x by 160-2x and combine like terms.

$$44800-880x+4x^{2}=16000$$

Subtract 16000 from both sides.

$$44800-880x+4x^{2}-16000=0$$

Subtract 16000 from 44800 to get 28800.

$$28800-880x+4x^{2}=0$$

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.

$$4x^{2}-880x+28800=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, -880 for b, and 28800 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-880\right)±\sqrt{\left(-880\right)^{2}-4\times 4\times 28800}}{2\times 4}$$

Square -880.

$$x=\frac{-\left(-880\right)±\sqrt{774400-4\times 4\times 28800}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-\left(-880\right)±\sqrt{774400-16\times 28800}}{2\times 4}$$

Multiply -16 times 28800.

$$x=\frac{-\left(-880\right)±\sqrt{774400-460800}}{2\times 4}$$

Add 774400 to -460800.

$$x=\frac{-\left(-880\right)±\sqrt{313600}}{2\times 4}$$

Take the square root of 313600.

$$x=\frac{-\left(-880\right)±560}{2\times 4}$$

The opposite of -880 is 880.

$$x=\frac{880±560}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{880±560}{8}$$

Now solve the equation x=\frac{880±560}{8} when ± is plus. Add 880 to 560.

$$x=\frac{1440}{8}$$

Divide 1440 by 8.

$$x=180$$

Now solve the equation x=\frac{880±560}{8} when ± is minus. Subtract 560 from 880.

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

Divide 320 by 8.

$$x=40$$

The equation is now solved.

$$x=180 x=40$$

Steps for Completing the Square

Use the distributive property to multiply 280-2x by 160-2x and combine like terms.

$$44800-880x+4x^{2}=16000$$

Subtract 44800 from both sides.

$$-880x+4x^{2}=16000-44800$$

Subtract 44800 from 16000 to get -28800.

$$-880x+4x^{2}=-28800$$

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.

$$4x^{2}-880x=-28800$$

Divide both sides by 4.

$$\frac{4x^{2}-880x}{4}=-\frac{28800}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$x^{2}+\left(-\frac{880}{4}\right)x=-\frac{28800}{4}$$

Divide -880 by 4.

$$x^{2}-220x=-\frac{28800}{4}$$

Divide -28800 by 4.

$$x^{2}-220x=-7200$$

Divide -220, the coefficient of the x term, by 2 to get -110. Then add the square of -110 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-220x+\left(-110\right)^{2}=-7200+\left(-110\right)^{2}$$

Square -110.

$$x^{2}-220x+12100=-7200+12100$$

Add -7200 to 12100.

$$x^{2}-220x+12100=4900$$

Factor x^{2}-220x+12100. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-110\right)^{2}=4900$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-110\right)^{2}}=\sqrt{4900}$$

Simplify.

$$x-110=70 x-110=-70$$

Add 110 to both sides of the equation.

$$x=180 x=40$$