Solve x^2-6x-72

$${ x }^{ 2 } -6x-72$$

Factor

$$\left(x-12\right)\left(x+6\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx-72. To find a and b, set up a system to be solved.

$$a+b=-6 ab=1\left(-72\right)=-72$$

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 -72.

$$1,-72 2,-36 3,-24 4,-18 6,-12 8,-9$$

Calculate the sum for each pair.

$$1-72=-71 2-36=-34 3-24=-21 4-18=-14 6-12=-6 8-9=-1$$

The solution is the pair that gives sum -6.

$$a=-12 b=6$$

Rewrite x^{2}-6x-72 as \left(x^{2}-12x\right)+\left(6x-72\right).

$$\left(x^{2}-12x\right)+\left(6x-72\right)$$

Factor out x in the first and 6 in the second group.

$$x\left(x-12\right)+6\left(x-12\right)$$

Factor out common term x-12 by using distributive property.

$$\left(x-12\right)\left(x+6\right)$$

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.

$$x^{2}-6x-72=0$$

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.

$$x=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\left(-72\right)}}{2}$$

Square -6.

$$x=\frac{-\left(-6\right)±\sqrt{36-4\left(-72\right)}}{2}$$

Multiply -4 times -72.

$$x=\frac{-\left(-6\right)±\sqrt{36+288}}{2}$$

Add 36 to 288.

$$x=\frac{-\left(-6\right)±\sqrt{324}}{2}$$

Take the square root of 324.

$$x=\frac{-\left(-6\right)±18}{2}$$

The opposite of -6 is 6.

$$x=\frac{6±18}{2}$$

Now solve the equation x=\frac{6±18}{2} when ± is plus. Add 6 to 18.

$$x=\frac{24}{2}$$

Divide 24 by 2.

$$x=12$$

Now solve the equation x=\frac{6±18}{2} when ± is minus. Subtract 18 from 6.

$$x=-\frac{12}{2}$$

Divide -12 by 2.

$$x=-6$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 12 for x_{1} and -6 for x_{2}.

$$x^{2}-6x-72=\left(x-12\right)\left(x-\left(-6\right)\right)$$

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

$$x^{2}-6x-72=\left(x-12\right)\left(x+6\right)$$

Evaluate

$$\left(x-12\right)\left(x+6\right)$$