Solve x^2-6x-16

$${ x }^{ 2 } -6x-16$$

Factor

$$\left(x-8\right)\left(x+2\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx-16. To find a and b, set up a system to be solved.

$$a+b=-6 ab=1\left(-16\right)=-16$$

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 -16.

$$1,-16 2,-8 4,-4$$

Calculate the sum for each pair.

$$1-16=-15 2-8=-6 4-4=0$$

The solution is the pair that gives sum -6.

$$a=-8 b=2$$

Rewrite x^{2}-6x-16 as \left(x^{2}-8x\right)+\left(2x-16\right).

$$\left(x^{2}-8x\right)+\left(2x-16\right)$$

Factor out x in the first and 2 in the second group.

$$x\left(x-8\right)+2\left(x-8\right)$$

Factor out common term x-8 by using distributive property.

$$\left(x-8\right)\left(x+2\right)$$

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.

$$x^{2}-6x-16=0$$

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.

$$x=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\left(-16\right)}}{2}$$

Square -6.

$$x=\frac{-\left(-6\right)±\sqrt{36-4\left(-16\right)}}{2}$$

Multiply -4 times -16.

$$x=\frac{-\left(-6\right)±\sqrt{36+64}}{2}$$

Add 36 to 64.

$$x=\frac{-\left(-6\right)±\sqrt{100}}{2}$$

Take the square root of 100.

$$x=\frac{-\left(-6\right)±10}{2}$$

The opposite of -6 is 6.

$$x=\frac{6±10}{2}$$

Now solve the equation x=\frac{6±10}{2} when ± is plus. Add 6 to 10.

$$x=\frac{16}{2}$$

Divide 16 by 2.

$$x=8$$

Now solve the equation x=\frac{6±10}{2} when ± is minus. Subtract 10 from 6.

$$x=-\frac{4}{2}$$

Divide -4 by 2.

$$x=-2$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 8 for x_{1} and -2 for x_{2}.

$$x^{2}-6x-16=\left(x-8\right)\left(x-\left(-2\right)\right)$$

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

$$x^{2}-6x-16=\left(x-8\right)\left(x+2\right)$$

Evaluate

$$\left(x-8\right)\left(x+2\right)$$