Solve (72-2x)(48-2x)=2160

$$(72-2x)(48-2x)=2160$$

Solve for x

$$x=54$$

$$x=6$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 72-2x by 48-2x and combine like terms.

$$3456-240x+4x^{2}=2160$$

Subtract 2160 from both sides.

$$3456-240x+4x^{2}-2160=0$$

Subtract 2160 from 3456 to get 1296.

$$1296-240x+4x^{2}=0$$

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.

$$4x^{2}-240x+1296=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, -240 for b, and 1296 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-240\right)±\sqrt{\left(-240\right)^{2}-4\times 4\times 1296}}{2\times 4}$$

Square -240.

$$x=\frac{-\left(-240\right)±\sqrt{57600-4\times 4\times 1296}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-\left(-240\right)±\sqrt{57600-16\times 1296}}{2\times 4}$$

Multiply -16 times 1296.

$$x=\frac{-\left(-240\right)±\sqrt{57600-20736}}{2\times 4}$$

Add 57600 to -20736.

$$x=\frac{-\left(-240\right)±\sqrt{36864}}{2\times 4}$$

Take the square root of 36864.

$$x=\frac{-\left(-240\right)±192}{2\times 4}$$

The opposite of -240 is 240.

$$x=\frac{240±192}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{240±192}{8}$$

Now solve the equation x=\frac{240±192}{8} when ± is plus. Add 240 to 192.

$$x=\frac{432}{8}$$

Divide 432 by 8.

$$x=54$$

Now solve the equation x=\frac{240±192}{8} when ± is minus. Subtract 192 from 240.

$$x=\frac{48}{8}$$

Divide 48 by 8.

$$x=6$$

The equation is now solved.

$$x=54 x=6$$

Steps for Completing the Square

Use the distributive property to multiply 72-2x by 48-2x and combine like terms.

$$3456-240x+4x^{2}=2160$$

Subtract 3456 from both sides.

$$-240x+4x^{2}=2160-3456$$

Subtract 3456 from 2160 to get -1296.

$$-240x+4x^{2}=-1296$$

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.

$$4x^{2}-240x=-1296$$

Divide both sides by 4.

$$\frac{4x^{2}-240x}{4}=-\frac{1296}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$x^{2}+\left(-\frac{240}{4}\right)x=-\frac{1296}{4}$$

Divide -240 by 4.

$$x^{2}-60x=-\frac{1296}{4}$$

Divide -1296 by 4.

$$x^{2}-60x=-324$$

Divide -60, the coefficient of the x term, by 2 to get -30. Then add the square of -30 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-60x+\left(-30\right)^{2}=-324+\left(-30\right)^{2}$$

Square -30.

$$x^{2}-60x+900=-324+900$$

Add -324 to 900.

$$x^{2}-60x+900=576$$

Factor x^{2}-60x+900. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-30\right)^{2}=576$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-30\right)^{2}}=\sqrt{576}$$

Simplify.

$$x-30=24 x-30=-24$$

Add 30 to both sides of the equation.

$$x=54 x=6$$