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

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

Solve for x

$$x=14$$

$$x=32$$

Steps Using the Quadratic Formula

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

$$1040-92x+2x^{2}=144$$

Subtract 144 from both sides.

$$1040-92x+2x^{2}-144=0$$

Subtract 144 from 1040 to get 896.

$$896-92x+2x^{2}=0$$

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.

$$2x^{2}-92x+896=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -92 for b, and 896 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-92\right)±\sqrt{\left(-92\right)^{2}-4\times 2\times 896}}{2\times 2}$$

Square -92.

$$x=\frac{-\left(-92\right)±\sqrt{8464-4\times 2\times 896}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-\left(-92\right)±\sqrt{8464-8\times 896}}{2\times 2}$$

Multiply -8 times 896.

$$x=\frac{-\left(-92\right)±\sqrt{8464-7168}}{2\times 2}$$

Add 8464 to -7168.

$$x=\frac{-\left(-92\right)±\sqrt{1296}}{2\times 2}$$

Take the square root of 1296.

$$x=\frac{-\left(-92\right)±36}{2\times 2}$$

The opposite of -92 is 92.

$$x=\frac{92±36}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{92±36}{4}$$

Now solve the equation x=\frac{92±36}{4} when ± is plus. Add 92 to 36.

$$x=\frac{128}{4}$$

Divide 128 by 4.

$$x=32$$

Now solve the equation x=\frac{92±36}{4} when ± is minus. Subtract 36 from 92.

$$x=\frac{56}{4}$$

Divide 56 by 4.

$$x=14$$

The equation is now solved.

$$x=32 x=14$$

Steps for Completing the Square

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

$$1040-92x+2x^{2}=144$$

Subtract 1040 from both sides.

$$-92x+2x^{2}=144-1040$$

Subtract 1040 from 144 to get -896.

$$-92x+2x^{2}=-896$$

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.

$$2x^{2}-92x=-896$$

Divide both sides by 2.

$$\frac{2x^{2}-92x}{2}=-\frac{896}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\left(-\frac{92}{2}\right)x=-\frac{896}{2}$$

Divide -92 by 2.

$$x^{2}-46x=-\frac{896}{2}$$

Divide -896 by 2.

$$x^{2}-46x=-448$$

Divide -46, the coefficient of the x term, by 2 to get -23. Then add the square of -23 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-46x+\left(-23\right)^{2}=-448+\left(-23\right)^{2}$$

Square -23.

$$x^{2}-46x+529=-448+529$$

Add -448 to 529.

$$x^{2}-46x+529=81$$

Factor x^{2}-46x+529. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-23\right)^{2}=81$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-23\right)^{2}}=\sqrt{81}$$

Simplify.

$$x-23=9 x-23=-9$$

Add 23 to both sides of the equation.

$$x=32 x=14$$