$\exponential{x}{2} + 11 x + 24$
Factor
Evaluate
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a+b=11 ab=1\times 24=24
Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx+24. To find a and b, set up a system to be solved.
1,24 2,12 3,8 4,6
Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 24.
1+24=25 2+12=14 3+8=11 4+6=10
Calculate the sum for each pair.
a=3 b=8
The solution is the pair that gives sum 11.
\left(x^{2}+3x\right)+\left(8x+24\right)
Rewrite x^{2}+11x+24 as \left(x^{2}+3x\right)+\left(8x+24\right).
x\left(x+3\right)+8\left(x+3\right)
Factor out x in the first and 8 in the second group.
\left(x+3\right)\left(x+8\right)
Factor out common term x+3 by using distributive property.
x^{2}+11x+24=0
Quadratic polynomial can be factored using the transformation ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), where x_{1} and x_{2} are the solutions of the quadratic equation ax^{2}+bx+c=0.
x=\frac{-11±\sqrt{11^{2}-4\times 24}}{2}
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=\frac{-11±\sqrt{121-4\times 24}}{2}
Square 11.
x=\frac{-11±\sqrt{121-96}}{2}
Multiply -4 times 24.
x=\frac{-11±\sqrt{25}}{2}
x=\frac{-11±5}{2}
Take the square root of 25.
x=\frac{-6}{2}
Now solve the equation x=\frac{-11±5}{2} when ± is plus. Add -11 to 5.
x=-3
Divide -6 by 2.
x=\frac{-16}{2}
Now solve the equation x=\frac{-11±5}{2} when ± is minus. Subtract 5 from -11.
x=-8
Divide -16 by 2.
x^{2}+11x+24=\left(x-\left(-3\right)\right)\left(x-\left(-8\right)\right)
Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -3 for x_{1} and -8 for x_{2}.
x^{2}+11x+24=\left(x+3\right)\left(x+8\right)
Simplify all the expressions of the form p-\left(-q\right) to p+q.