Solve for x
x=-18
x=13
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x^{2}+5x+6-240=0
Use the distributive property to multiply x+3 by x+2 and combine like terms.
x^{2}+5x-234=0
Subtract 240 from 6 to get -234.
a+b=5 ab=-234
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.
-1,234 -2,117 -3,78 -6,39 -9,26 -13,18
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=233 -2+117=115 -3+78=75 -6+39=33 -9+26=17 -13+18=5
Calculate the sum for each pair.
a=-13 b=18
The solution is the pair that gives sum 5.
\left(x-13\right)\left(x+18\right)
Rewrite factored expression \left(x+a\right)\left(x+b\right) using the obtained values.
x=13 x=-18
To find equation solutions, solve x-13=0 and x+18=0.
x^{2}+5x+6-240=0
Use the distributive property to multiply x+3 by x+2 and combine like terms.
x^{2}+5x-234=0
Subtract 240 from 6 to get -234.
a+b=5 ab=1\left(-234\right)=-234
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.
-1,234 -2,117 -3,78 -6,39 -9,26 -13,18
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=233 -2+117=115 -3+78=75 -6+39=33 -9+26=17 -13+18=5
Calculate the sum for each pair.
a=-13 b=18
The solution is the pair that gives sum 5.
\left(x^{2}-13x\right)+\left(18x-234\right)
Rewrite x^{2}+5x-234 as \left(x^{2}-13x\right)+\left(18x-234\right).
x\left(x-13\right)+18\left(x-13\right)
Factor out x in the first and 18 in the second group.
\left(x-13\right)\left(x+18\right)
Factor out common term x-13 by using distributive property.
x=13 x=-18
To find equation solutions, solve x-13=0 and x+18=0.
x^{2}+5x+6-240=0
Use the distributive property to multiply x+3 by x+2 and combine like terms.
x^{2}+5x-234=0
Subtract 240 from 6 to get -234.
x=\frac{-5±\sqrt{5^{2}-4\left(-234\right)}}{2}
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{25-4\left(-234\right)}}{2}
Square 5.
x=\frac{-5±\sqrt{25+936}}{2}
Multiply -4 times -234.
x=\frac{-5±\sqrt{961}}{2}
Add 25 to 936.
x=\frac{-5±31}{2}
Take the square root of 961.
x=\frac{26}{2}
Now solve the equation x=\frac{-5±31}{2} when ± is plus. Add -5 to 31.
x=13
Divide 26 by 2.
x=-\frac{36}{2}
Now solve the equation x=\frac{-5±31}{2} when ± is minus. Subtract 31 from -5.
x=-18
Divide -36 by 2.
x=13 x=-18
The equation is now solved.
x^{2}+5x+6-240=0
Use the distributive property to multiply x+3 by x+2 and combine like terms.
x^{2}+5x-234=0
Subtract 240 from 6 to get -234.
x^{2}+5x=234
Add 234 to both sides. Anything plus zero gives itself.
x^{2}+5x+\left(\frac{5}{2}\right)^{2}=234+\left(\frac{5}{2}\right)^{2}
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+\frac{25}{4}=234+\frac{25}{4}
Square \frac{5}{2} by squaring both the numerator and the denominator of the fraction.
x^{2}+5x+\frac{25}{4}=\frac{961}{4}
Add 234 to \frac{25}{4}.
\left(x+\frac{5}{2}\right)^{2}=\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}.
\sqrt{\left(x+\frac{5}{2}\right)^{2}}=\sqrt{\frac{961}{4}}
Take the square root of both sides of the equation.
x+\frac{5}{2}=\frac{31}{2} x+\frac{5}{2}=-\frac{31}{2}
Simplify.
x=13 x=-18
Subtract \frac{5}{2} from both sides of the equation.
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