Solve (32-2x)(20-x)=57

$$( 32 - 2 x ) ( 20 - x ) = 57$$

Solve for x

$$x=\frac{\sqrt{130}}{2}+18\approx 23.700877125$$

$$x=-\frac{\sqrt{130}}{2}+18\approx 12.299122875$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 32-2x by 20-x and combine like terms.

$$640-72x+2x^{2}=57$$

Subtract 57 from both sides.

$$640-72x+2x^{2}-57=0$$

Subtract 57 from 640 to get 583.

$$583-72x+2x^{2}=0$$

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.

$$2x^{2}-72x+583=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -72 for b, and 583 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-72\right)±\sqrt{\left(-72\right)^{2}-4\times 2\times 583}}{2\times 2}$$

Square -72.

$$x=\frac{-\left(-72\right)±\sqrt{5184-4\times 2\times 583}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-\left(-72\right)±\sqrt{5184-8\times 583}}{2\times 2}$$

Multiply -8 times 583.

$$x=\frac{-\left(-72\right)±\sqrt{5184-4664}}{2\times 2}$$

Add 5184 to -4664.

$$x=\frac{-\left(-72\right)±\sqrt{520}}{2\times 2}$$

Take the square root of 520.

$$x=\frac{-\left(-72\right)±2\sqrt{130}}{2\times 2}$$

The opposite of -72 is 72.

$$x=\frac{72±2\sqrt{130}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{72±2\sqrt{130}}{4}$$

Now solve the equation x=\frac{72±2\sqrt{130}}{4} when ± is plus. Add 72 to 2\sqrt{130}.

$$x=\frac{2\sqrt{130}+72}{4}$$

Divide 72+2\sqrt{130} by 4.

$$x=\frac{\sqrt{130}}{2}+18$$

Now solve the equation x=\frac{72±2\sqrt{130}}{4} when ± is minus. Subtract 2\sqrt{130} from 72.

$$x=\frac{72-2\sqrt{130}}{4}$$

Divide 72-2\sqrt{130} by 4.

$$x=-\frac{\sqrt{130}}{2}+18$$

The equation is now solved.

$$x=\frac{\sqrt{130}}{2}+18 x=-\frac{\sqrt{130}}{2}+18$$

Steps for Completing the Square

Use the distributive property to multiply 32-2x by 20-x and combine like terms.

$$640-72x+2x^{2}=57$$

Subtract 640 from both sides.

$$-72x+2x^{2}=57-640$$

Subtract 640 from 57 to get -583.

$$-72x+2x^{2}=-583$$

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.

$$2x^{2}-72x=-583$$

Divide both sides by 2.

$$\frac{2x^{2}-72x}{2}=-\frac{583}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\left(-\frac{72}{2}\right)x=-\frac{583}{2}$$

Divide -72 by 2.

$$x^{2}-36x=-\frac{583}{2}$$

Divide -36, the coefficient of the x term, by 2 to get -18. Then add the square of -18 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-36x+\left(-18\right)^{2}=-\frac{583}{2}+\left(-18\right)^{2}$$

Square -18.

$$x^{2}-36x+324=-\frac{583}{2}+324$$

Add -\frac{583}{2} to 324.

$$x^{2}-36x+324=\frac{65}{2}$$

Factor x^{2}-36x+324. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-18\right)^{2}=\frac{65}{2}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-18\right)^{2}}=\sqrt{\frac{65}{2}}$$

Simplify.

$$x-18=\frac{\sqrt{130}}{2} x-18=-\frac{\sqrt{130}}{2}$$

Add 18 to both sides of the equation.

$$x=\frac{\sqrt{130}}{2}+18 x=-\frac{\sqrt{130}}{2}+18$$