Solve (20-x)(32-x)=540

$$(20-x)(32-x)=540$$

Solve for x

$$x=50$$

$$x=2$$

Steps Using the Quadratic Formula

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

$$640-52x+x^{2}=540$$

Subtract 540 from both sides.

$$640-52x+x^{2}-540=0$$

Subtract 540 from 640 to get 100.

$$100-52x+x^{2}=0$$

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.

$$x^{2}-52x+100=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -52 for b, and 100 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-52\right)±\sqrt{\left(-52\right)^{2}-4\times 100}}{2}$$

Square -52.

$$x=\frac{-\left(-52\right)±\sqrt{2704-4\times 100}}{2}$$

Multiply -4 times 100.

$$x=\frac{-\left(-52\right)±\sqrt{2704-400}}{2}$$

Add 2704 to -400.

$$x=\frac{-\left(-52\right)±\sqrt{2304}}{2}$$

Take the square root of 2304.

$$x=\frac{-\left(-52\right)±48}{2}$$

The opposite of -52 is 52.

$$x=\frac{52±48}{2}$$

Now solve the equation x=\frac{52±48}{2} when ± is plus. Add 52 to 48.

$$x=\frac{100}{2}$$

Divide 100 by 2.

$$x=50$$

Now solve the equation x=\frac{52±48}{2} when ± is minus. Subtract 48 from 52.

$$x=\frac{4}{2}$$

Divide 4 by 2.

$$x=2$$

The equation is now solved.

$$x=50 x=2$$

Steps for Completing the Square

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

$$640-52x+x^{2}=540$$

Subtract 640 from both sides.

$$-52x+x^{2}=540-640$$

Subtract 640 from 540 to get -100.

$$-52x+x^{2}=-100$$

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.

$$x^{2}-52x=-100$$

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.

$$x^{2}-52x+\left(-26\right)^{2}=-100+\left(-26\right)^{2}$$

Square -26.

$$x^{2}-52x+676=-100+676$$

Add -100 to 676.

$$x^{2}-52x+676=576$$

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}.

$$\left(x-26\right)^{2}=576$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-26\right)^{2}}=\sqrt{576}$$

Simplify.

$$x-26=24 x-26=-24$$

Add 26 to both sides of the equation.

$$x=50 x=2$$