$\exponential{x}{2} - 7 x + 12$
Factor
Evaluate
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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 negative, a and b are both negative. 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=-4 b=-3
The solution is the pair that gives sum -7.
\left(x^{2}-4x\right)+\left(-3x+12\right)
Rewrite x^{2}-7x+12 as \left(x^{2}-4x\right)+\left(-3x+12\right).
x\left(x-4\right)-3\left(x-4\right)
Factor out x in the first and -3 in the second group.
\left(x-4\right)\left(x-3\right)
Factor out common term x-4 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{-\left(-7\right)±\sqrt{\left(-7\right)^{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{-\left(-7\right)±\sqrt{49-4\times 12}}{2}
Square -7.
x=\frac{-\left(-7\right)±\sqrt{49-48}}{2}
Multiply -4 times 12.
x=\frac{-\left(-7\right)±\sqrt{1}}{2}
x=\frac{-\left(-7\right)±1}{2}
Take the square root of 1.
x=\frac{7±1}{2}
The opposite of -7 is 7.
x=\frac{8}{2}
Now solve the equation x=\frac{7±1}{2} when ± is plus. Add 7 to 1.
x=4
Divide 8 by 2.
x=\frac{6}{2}
Now solve the equation x=\frac{7±1}{2} when ± is minus. Subtract 1 from 7.
x=3
Divide 6 by 2.
x^{2}-7x+12=\left(x-4\right)\left(x-3\right)
Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 4 for x_{1} and 3 for x_{2}.