Solve (x+3)(x+2)-240=0

$$(x+3)(x+2)-240=0$$

Solve for x

$$x=-18$$

$$x=13$$

Steps Using Factoring

Use the distributive property to multiply x+3 by x+2 and combine like terms.

$$x^{2}+5x+6-240=0$$

Subtract 240 from 6 to get -234.

$$x^{2}+5x-234=0$$

To solve the equation, factor x^{2}+5x-234 using formula x^{2}+\left(a+b\right)x+ab=\left(x+a\right)\left(x+b\right). To find a and b, set up a system to be solved.

$$a+b=5 ab=-234$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -234.

$$-1,234 -2,117 -3,78 -6,39 -9,26 -13,18$$

Calculate the sum for each pair.

$$-1+234=233 -2+117=115 -3+78=75 -6+39=33 -9+26=17 -13+18=5$$

The solution is the pair that gives sum 5.

$$a=-13 b=18$$

Rewrite factored expression \left(x+a\right)\left(x+b\right) using the obtained values.

$$\left(x-13\right)\left(x+18\right)$$

To find equation solutions, solve x-13=0 and x+18=0.

$$x=13 x=-18$$

Steps Using Factoring By Grouping

Use the distributive property to multiply x+3 by x+2 and combine like terms.

$$x^{2}+5x+6-240=0$$

Subtract 240 from 6 to get -234.

$$x^{2}+5x-234=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx-234. To find a and b, set up a system to be solved.

$$a+b=5 ab=1\left(-234\right)=-234$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -234.

$$-1,234 -2,117 -3,78 -6,39 -9,26 -13,18$$

Calculate the sum for each pair.

$$-1+234=233 -2+117=115 -3+78=75 -6+39=33 -9+26=17 -13+18=5$$

The solution is the pair that gives sum 5.

$$a=-13 b=18$$

Rewrite x^{2}+5x-234 as \left(x^{2}-13x\right)+\left(18x-234\right).

$$\left(x^{2}-13x\right)+\left(18x-234\right)$$

Factor out x in the first and 18 in the second group.

$$x\left(x-13\right)+18\left(x-13\right)$$

Factor out common term x-13 by using distributive property.

$$\left(x-13\right)\left(x+18\right)$$

To find equation solutions, solve x-13=0 and x+18=0.

$$x=13 x=-18$$

Steps Using the Quadratic Formula

Use the distributive property to multiply x+3 by x+2 and combine like terms.

$$x^{2}+5x+6-240=0$$

Subtract 240 from 6 to get -234.

$$x^{2}+5x-234=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 5 for b, and -234 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-5±\sqrt{5^{2}-4\left(-234\right)}}{2}$$

Square 5.

$$x=\frac{-5±\sqrt{25-4\left(-234\right)}}{2}$$

Multiply -4 times -234.

$$x=\frac{-5±\sqrt{25+936}}{2}$$

Add 25 to 936.

$$x=\frac{-5±\sqrt{961}}{2}$$

Take the square root of 961.

$$x=\frac{-5±31}{2}$$

Now solve the equation x=\frac{-5±31}{2} when ± is plus. Add -5 to 31.

$$x=\frac{26}{2}$$

Divide 26 by 2.

$$x=13$$

Now solve the equation x=\frac{-5±31}{2} when ± is minus. Subtract 31 from -5.

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

Divide -36 by 2.

$$x=-18$$

The equation is now solved.

$$x=13 x=-18$$

Steps for Completing the Square

Use the distributive property to multiply x+3 by x+2 and combine like terms.

$$x^{2}+5x+6-240=0$$

Subtract 240 from 6 to get -234.

$$x^{2}+5x-234=0$$

Add 234 to both sides. Anything plus zero gives itself.

$$x^{2}+5x=234$$

Divide 5, the coefficient of the x term, by 2 to get \frac{5}{2}. Then add the square of \frac{5}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+5x+\left(\frac{5}{2}\right)^{2}=234+\left(\frac{5}{2}\right)^{2}$$

Square \frac{5}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+5x+\frac{25}{4}=234+\frac{25}{4}$$

Add 234 to \frac{25}{4}.

$$x^{2}+5x+\frac{25}{4}=\frac{961}{4}$$

Factor x^{2}+5x+\frac{25}{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}.

$$\left(x+\frac{5}{2}\right)^{2}=\frac{961}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{5}{2}\right)^{2}}=\sqrt{\frac{961}{4}}$$

Simplify.

$$x+\frac{5}{2}=\frac{31}{2} x+\frac{5}{2}=-\frac{31}{2}$$

Subtract \frac{5}{2} from both sides of the equation.

$$x=13 x=-18$$