Factor
\left(x-9\right)\left(x+3\right)
Evaluate
\left(x-9\right)\left(x+3\right)
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a+b=-6 ab=1\left(-27\right)=-27
Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx-27. To find a and b, set up a system to be solved.
1,-27 3,-9
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 -27.
1-27=-26 3-9=-6
Calculate the sum for each pair.
a=-9 b=3
The solution is the pair that gives sum -6.
\left(x^{2}-9x\right)+\left(3x-27\right)
Rewrite x^{2}-6x-27 as \left(x^{2}-9x\right)+\left(3x-27\right).
x\left(x-9\right)+3\left(x-9\right)
Factor out x in the first and 3 in the second group.
\left(x-9\right)\left(x+3\right)
Factor out common term x-9 by using distributive property.
x^{2}-6x-27=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(-6\right)±\sqrt{\left(-6\right)^{2}-4\left(-27\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(-6\right)±\sqrt{36-4\left(-27\right)}}{2}
Square -6.
x=\frac{-\left(-6\right)±\sqrt{36+108}}{2}
Multiply -4 times -27.
x=\frac{-\left(-6\right)±\sqrt{144}}{2}
Add 36 to 108.
x=\frac{-\left(-6\right)±12}{2}
Take the square root of 144.
x=\frac{6±12}{2}
The opposite of -6 is 6.
x=\frac{18}{2}
Now solve the equation x=\frac{6±12}{2} when ± is plus. Add 6 to 12.
x=9
Divide 18 by 2.
x=-\frac{6}{2}
Now solve the equation x=\frac{6±12}{2} when ± is minus. Subtract 12 from 6.
x=-3
Divide -6 by 2.
x^{2}-6x-27=\left(x-9\right)\left(x-\left(-3\right)\right)
Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 9 for x_{1} and -3 for x_{2}.
x^{2}-6x-27=\left(x-9\right)\left(x+3\right)
Simplify all the expressions of the form p-\left(-q\right) to p+q.
Examples
Quadratic equation
{ x } ^ { 2 } - 4 x - 5 = 0
Trigonometry
4 \sin \theta \cos \theta = 2 \sin \theta
Linear equation
y = 3x + 4
Arithmetic
699 * 533
Matrix
\left[ \begin{array} { l l } { 2 } & { 3 } \\ { 5 } & { 4 } \end{array} \right] \left[ \begin{array} { l l l } { 2 } & { 0 } & { 3 } \\ { -1 } & { 1 } & { 5 } \end{array} \right]
Simultaneous equation
\left. \begin{cases} { 8x+2y = 46 } \\ { 7x+3y = 47 } \end{cases} \right.
Differentiation
\frac { d } { d x } \frac { ( 3 x ^ { 2 } - 2 ) } { ( x - 5 ) }
Integration
\int _ { 0 } ^ { 1 } x e ^ { - x ^ { 2 } } d x
Limits
\lim _{x \rightarrow-3} \frac{x^{2}-9}{x^{2}+2 x-3}