Solve x^2-6x-135

$${ x }^{ 2 } -6x-135$$

Factor

$$\left(x-15\right)\left(x+9\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx-135. To find a and b, set up a system to be solved.

$$a+b=-6 ab=1\left(-135\right)=-135$$

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 -135.

$$1,-135 3,-45 5,-27 9,-15$$

Calculate the sum for each pair.

$$1-135=-134 3-45=-42 5-27=-22 9-15=-6$$

The solution is the pair that gives sum -6.

$$a=-15 b=9$$

Rewrite x^{2}-6x-135 as \left(x^{2}-15x\right)+\left(9x-135\right).

$$\left(x^{2}-15x\right)+\left(9x-135\right)$$

Factor out x in the first and 9 in the second group.

$$x\left(x-15\right)+9\left(x-15\right)$$

Factor out common term x-15 by using distributive property.

$$\left(x-15\right)\left(x+9\right)$$

Steps Using the Quadratic Formula

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.

$$x^{2}-6x-135=0$$

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.

$$x=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\left(-135\right)}}{2}$$

Square -6.

$$x=\frac{-\left(-6\right)±\sqrt{36-4\left(-135\right)}}{2}$$

Multiply -4 times -135.

$$x=\frac{-\left(-6\right)±\sqrt{36+540}}{2}$$

Add 36 to 540.

$$x=\frac{-\left(-6\right)±\sqrt{576}}{2}$$

Take the square root of 576.

$$x=\frac{-\left(-6\right)±24}{2}$$

The opposite of -6 is 6.

$$x=\frac{6±24}{2}$$

Now solve the equation x=\frac{6±24}{2} when ± is plus. Add 6 to 24.

$$x=\frac{30}{2}$$

Divide 30 by 2.

$$x=15$$

Now solve the equation x=\frac{6±24}{2} when ± is minus. Subtract 24 from 6.

$$x=-\frac{18}{2}$$

Divide -18 by 2.

$$x=-9$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 15 for x_{1} and -9 for x_{2}.

$$x^{2}-6x-135=\left(x-15\right)\left(x-\left(-9\right)\right)$$

Simplify all the expressions of the form p-\left(-q\right) to p+q.

$$x^{2}-6x-135=\left(x-15\right)\left(x+9\right)$$

Evaluate

$$\left(x-15\right)\left(x+9\right)$$