Solve (32-x)*(20-x)=570

$$(32-x) \times (20-x)=570$$

Solve for x

$$x=26-\sqrt{606}\approx 1.38293275$$

$$x=\sqrt{606}+26\approx 50.61706725$$

Steps Using the Quadratic Formula

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

$$640-52x+x^{2}=570$$

Subtract 570 from both sides.

$$640-52x+x^{2}-570=0$$

Subtract 570 from 640 to get 70.

$$70-52x+x^{2}=0$$

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.

$$x^{2}-52x+70=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -52 for b, and 70 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-52\right)±\sqrt{\left(-52\right)^{2}-4\times 70}}{2}$$

Square -52.

$$x=\frac{-\left(-52\right)±\sqrt{2704-4\times 70}}{2}$$

Multiply -4 times 70.

$$x=\frac{-\left(-52\right)±\sqrt{2704-280}}{2}$$

Add 2704 to -280.

$$x=\frac{-\left(-52\right)±\sqrt{2424}}{2}$$

Take the square root of 2424.

$$x=\frac{-\left(-52\right)±2\sqrt{606}}{2}$$

The opposite of -52 is 52.

$$x=\frac{52±2\sqrt{606}}{2}$$

Now solve the equation x=\frac{52±2\sqrt{606}}{2} when ± is plus. Add 52 to 2\sqrt{606}.

$$x=\frac{2\sqrt{606}+52}{2}$$

Divide 52+2\sqrt{606} by 2.

$$x=\sqrt{606}+26$$

Now solve the equation x=\frac{52±2\sqrt{606}}{2} when ± is minus. Subtract 2\sqrt{606} from 52.

$$x=\frac{52-2\sqrt{606}}{2}$$

Divide 52-2\sqrt{606} by 2.

$$x=26-\sqrt{606}$$

The equation is now solved.

$$x=\sqrt{606}+26 x=26-\sqrt{606}$$

Steps for Completing the Square

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

$$640-52x+x^{2}=570$$

Subtract 640 from both sides.

$$-52x+x^{2}=570-640$$

Subtract 640 from 570 to get -70.

$$-52x+x^{2}=-70$$

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.

$$x^{2}-52x=-70$$

Divide -52, the coefficient of the x term, by 2 to get -26. Then add the square of -26 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-52x+\left(-26\right)^{2}=-70+\left(-26\right)^{2}$$

Square -26.

$$x^{2}-52x+676=-70+676$$

Add -70 to 676.

$$x^{2}-52x+676=606$$

Factor x^{2}-52x+676. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-26\right)^{2}=606$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-26\right)^{2}}=\sqrt{606}$$

Simplify.

$$x-26=\sqrt{606} x-26=-\sqrt{606}$$

Add 26 to both sides of the equation.

$$x=\sqrt{606}+26 x=26-\sqrt{606}$$