Solve (40-2x)(26-2x)=144

$$( 40 - 2 x ) ( 26 - 2 x ) = 144$$

Solve for x

$$x = \frac{33 - \sqrt{193}}{2} \approx 9.553778005$$

$$x = \frac{\sqrt{193} + 33}{2} \approx 23.446221995$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 40-2x by 26-2x and combine like terms.

$$1040-132x+4x^{2}=144$$

Subtract 144 from both sides.

$$1040-132x+4x^{2}-144=0$$

Subtract 144 from 1040 to get 896.

$$896-132x+4x^{2}=0$$

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.

$$4x^{2}-132x+896=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, -132 for b, and 896 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-132\right)±\sqrt{\left(-132\right)^{2}-4\times 4\times 896}}{2\times 4}$$

Square -132.

$$x=\frac{-\left(-132\right)±\sqrt{17424-4\times 4\times 896}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-\left(-132\right)±\sqrt{17424-16\times 896}}{2\times 4}$$

Multiply -16 times 896.

$$x=\frac{-\left(-132\right)±\sqrt{17424-14336}}{2\times 4}$$

Add 17424 to -14336.

$$x=\frac{-\left(-132\right)±\sqrt{3088}}{2\times 4}$$

Take the square root of 3088.

$$x=\frac{-\left(-132\right)±4\sqrt{193}}{2\times 4}$$

The opposite of -132 is 132.

$$x=\frac{132±4\sqrt{193}}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{132±4\sqrt{193}}{8}$$

Now solve the equation x=\frac{132±4\sqrt{193}}{8} when ± is plus. Add 132 to 4\sqrt{193}.

$$x=\frac{4\sqrt{193}+132}{8}$$

Divide 132+4\sqrt{193} by 8.

$$x=\frac{\sqrt{193}+33}{2}$$

Now solve the equation x=\frac{132±4\sqrt{193}}{8} when ± is minus. Subtract 4\sqrt{193} from 132.

$$x=\frac{132-4\sqrt{193}}{8}$$

Divide 132-4\sqrt{193} by 8.

$$x=\frac{33-\sqrt{193}}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{193}+33}{2} x=\frac{33-\sqrt{193}}{2}$$

Steps for Completing the Square

Use the distributive property to multiply 40-2x by 26-2x and combine like terms.

$$1040-132x+4x^{2}=144$$

Subtract 1040 from both sides.

$$-132x+4x^{2}=144-1040$$

Subtract 1040 from 144 to get -896.

$$-132x+4x^{2}=-896$$

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.

$$4x^{2}-132x=-896$$

Divide both sides by 4.

$$\frac{4x^{2}-132x}{4}=-\frac{896}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$x^{2}+\left(-\frac{132}{4}\right)x=-\frac{896}{4}$$

Divide -132 by 4.

$$x^{2}-33x=-\frac{896}{4}$$

Divide -896 by 4.

$$x^{2}-33x=-224$$

Divide -33, the coefficient of the x term, by 2 to get -\frac{33}{2}. Then add the square of -\frac{33}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-33x+\left(-\frac{33}{2}\right)^{2}=-224+\left(-\frac{33}{2}\right)^{2}$$

Square -\frac{33}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-33x+\frac{1089}{4}=-224+\frac{1089}{4}$$

Add -224 to \frac{1089}{4}.

$$x^{2}-33x+\frac{1089}{4}=\frac{193}{4}$$

Factor x^{2}-33x+\frac{1089}{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}.

$$\left(x-\frac{33}{2}\right)^{2}=\frac{193}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{33}{2}\right)^{2}}=\sqrt{\frac{193}{4}}$$

Simplify.

$$x-\frac{33}{2}=\frac{\sqrt{193}}{2} x-\frac{33}{2}=-\frac{\sqrt{193}}{2}$$

Add \frac{33}{2} to both sides of the equation.

$$x=\frac{\sqrt{193}+33}{2} x=\frac{33-\sqrt{193}}{2}$$