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a+b=7 ab=1\times 12=12
Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx+12. To find a and b, set up a system to be solved.
1,12 2,6 3,4
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 12.
1+12=13 2+6=8 3+4=7
Calculate the sum for each pair.
a=3 b=4
The solution is the pair that gives sum 7.
\left(x^{2}+3x\right)+\left(4x+12\right)
Rewrite x^{2}+7x+12 as \left(x^{2}+3x\right)+\left(4x+12\right).
x\left(x+3\right)+4\left(x+3\right)
Factor out x in the first and 4 in the second group.
\left(x+3\right)\left(x+4\right)
Factor out common term x+3 by using distributive property.
x^{2}+7x+12=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{-7±\sqrt{7^{2}-4\times 12}}{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{-7±\sqrt{49-4\times 12}}{2}
Square 7.
x=\frac{-7±\sqrt{49-48}}{2}
Multiply -4 times 12.
x=\frac{-7±\sqrt{1}}{2}
Add 49 to -48.
x=\frac{-7±1}{2}
Take the square root of 1.
x=-\frac{6}{2}
Now solve the equation x=\frac{-7±1}{2} when ± is plus. Add -7 to 1.
x=-3
Divide -6 by 2.
x=-\frac{8}{2}
Now solve the equation x=\frac{-7±1}{2} when ± is minus. Subtract 1 from -7.
x=-4
Divide -8 by 2.
x^{2}+7x+12=\left(x-\left(-3\right)\right)\left(x-\left(-4\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 -4 for x_{2}.
x^{2}+7x+12=\left(x+3\right)\left(x+4\right)
Simplify all the expressions of the form p-\left(-q\right) to p+q.