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x^{2}-x-6
Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.
a+b=-1 ab=1\left(-6\right)=-6
Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx-6. To find a and b, set up a system to be solved.
1,-6 2,-3
Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -6.
1-6=-5 2-3=-1
Calculate the sum for each pair.
a=-3 b=2
The solution is the pair that gives sum -1.
\left(x^{2}-3x\right)+\left(2x-6\right)
Rewrite x^{2}-x-6 as \left(x^{2}-3x\right)+\left(2x-6\right).
x\left(x-3\right)+2\left(x-3\right)
Factor out x in the first and 2 in the second group.
\left(x-3\right)\left(x+2\right)
Factor out common term x-3 by using distributive property.
x^{2}-x-6=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(-1\right)±\sqrt{1-4\left(-6\right)}}{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(-1\right)±\sqrt{1+24}}{2}
Multiply -4 times -6.
x=\frac{-\left(-1\right)±\sqrt{25}}{2}
Add 1 to 24.
x=\frac{-\left(-1\right)±5}{2}
Take the square root of 25.
x=\frac{1±5}{2}
The opposite of -1 is 1.
x=\frac{6}{2}
Now solve the equation x=\frac{1±5}{2} when ± is plus. Add 1 to 5.
x=3
Divide 6 by 2.
x=-\frac{4}{2}
Now solve the equation x=\frac{1±5}{2} when ± is minus. Subtract 5 from 1.
x=-2
Divide -4 by 2.
x^{2}-x-6=\left(x-3\right)\left(x-\left(-2\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 -2 for x_{2}.
x^{2}-x-6=\left(x-3\right)\left(x+2\right)
Simplify all the expressions of the form p-\left(-q\right) to p+q.