Solve left(20-xright)left(32-2xright)=570

$$\left( 20-x \right) \left( 32-2x \right) = 570$$

Solve for x

$$x=1$$

$$x=35$$

Steps Using the Quadratic Formula

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

$$640-72x+2x^{2}=570$$

Subtract 570 from both sides.

$$640-72x+2x^{2}-570=0$$

Subtract 570 from 640 to get 70.

$$70-72x+2x^{2}=0$$

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.

$$2x^{2}-72x+70=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -72 for b, and 70 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-72\right)±\sqrt{\left(-72\right)^{2}-4\times 2\times 70}}{2\times 2}$$

Square -72.

$$x=\frac{-\left(-72\right)±\sqrt{5184-4\times 2\times 70}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-\left(-72\right)±\sqrt{5184-8\times 70}}{2\times 2}$$

Multiply -8 times 70.

$$x=\frac{-\left(-72\right)±\sqrt{5184-560}}{2\times 2}$$

Add 5184 to -560.

$$x=\frac{-\left(-72\right)±\sqrt{4624}}{2\times 2}$$

Take the square root of 4624.

$$x=\frac{-\left(-72\right)±68}{2\times 2}$$

The opposite of -72 is 72.

$$x=\frac{72±68}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{72±68}{4}$$

Now solve the equation x=\frac{72±68}{4} when ± is plus. Add 72 to 68.

$$x=\frac{140}{4}$$

Divide 140 by 4.

$$x=35$$

Now solve the equation x=\frac{72±68}{4} when ± is minus. Subtract 68 from 72.

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

Divide 4 by 4.

$$x=1$$

The equation is now solved.

$$x=35 x=1$$

Steps for Completing the Square

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

$$640-72x+2x^{2}=570$$

Subtract 640 from both sides.

$$-72x+2x^{2}=570-640$$

Subtract 640 from 570 to get -70.

$$-72x+2x^{2}=-70$$

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.

$$2x^{2}-72x=-70$$

Divide both sides by 2.

$$\frac{2x^{2}-72x}{2}=-\frac{70}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\left(-\frac{72}{2}\right)x=-\frac{70}{2}$$

Divide -72 by 2.

$$x^{2}-36x=-\frac{70}{2}$$

Divide -70 by 2.

$$x^{2}-36x=-35$$

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.

$$x^{2}-36x+\left(-18\right)^{2}=-35+\left(-18\right)^{2}$$

Square -18.

$$x^{2}-36x+324=-35+324$$

Add -35 to 324.

$$x^{2}-36x+324=289$$

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}.

$$\left(x-18\right)^{2}=289$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-18\right)^{2}}=\sqrt{289}$$

Simplify.

$$x-18=17 x-18=-17$$

Add 18 to both sides of the equation.

$$x=35 x=1$$