Solve (18-x)(13-x)=66

$$(18-x)(13-x)=66$$

Solve for x

$$x=24$$

$$x=7$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 18-x by 13-x and combine like terms.

$$234-31x+x^{2}=66$$

Subtract 66 from both sides.

$$234-31x+x^{2}-66=0$$

Subtract 66 from 234 to get 168.

$$168-31x+x^{2}=0$$

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.

$$x^{2}-31x+168=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -31 for b, and 168 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-31\right)±\sqrt{\left(-31\right)^{2}-4\times 168}}{2}$$

Square -31.

$$x=\frac{-\left(-31\right)±\sqrt{961-4\times 168}}{2}$$

Multiply -4 times 168.

$$x=\frac{-\left(-31\right)±\sqrt{961-672}}{2}$$

Add 961 to -672.

$$x=\frac{-\left(-31\right)±\sqrt{289}}{2}$$

Take the square root of 289.

$$x=\frac{-\left(-31\right)±17}{2}$$

The opposite of -31 is 31.

$$x=\frac{31±17}{2}$$

Now solve the equation x=\frac{31±17}{2} when ± is plus. Add 31 to 17.

$$x=\frac{48}{2}$$

Divide 48 by 2.

$$x=24$$

Now solve the equation x=\frac{31±17}{2} when ± is minus. Subtract 17 from 31.

$$x=\frac{14}{2}$$

Divide 14 by 2.

$$x=7$$

The equation is now solved.

$$x=24 x=7$$

Steps for Completing the Square

Use the distributive property to multiply 18-x by 13-x and combine like terms.

$$234-31x+x^{2}=66$$

Subtract 234 from both sides.

$$-31x+x^{2}=66-234$$

Subtract 234 from 66 to get -168.

$$-31x+x^{2}=-168$$

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.

$$x^{2}-31x=-168$$

Divide -31, the coefficient of the x term, by 2 to get -\frac{31}{2}. Then add the square of -\frac{31}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-31x+\left(-\frac{31}{2}\right)^{2}=-168+\left(-\frac{31}{2}\right)^{2}$$

Square -\frac{31}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-31x+\frac{961}{4}=-168+\frac{961}{4}$$

Add -168 to \frac{961}{4}.

$$x^{2}-31x+\frac{961}{4}=\frac{289}{4}$$

Factor x^{2}-31x+\frac{961}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{31}{2}\right)^{2}=\frac{289}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{31}{2}\right)^{2}}=\sqrt{\frac{289}{4}}$$

Simplify.

$$x-\frac{31}{2}=\frac{17}{2} x-\frac{31}{2}=-\frac{17}{2}$$

Add \frac{31}{2} to both sides of the equation.

$$x=24 x=7$$