Solve x^2+10x+16

$${ x }^{ 2 } +10x+16$$

Factor

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

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.

$$a+b=10 ab=1\times 16=16$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 16.

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

Calculate the sum for each pair.

$$1+16=17 2+8=10 4+4=8$$

The solution is the pair that gives sum 10.

$$a=2 b=8$$

Rewrite x^{2}+10x+16 as \left(x^{2}+2x\right)+\left(8x+16\right).

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

Factor out x in the first and 8 in the second group.

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

Factor out common term x+2 by using distributive property.

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

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.

$$x^{2}+10x+16=0$$

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.

$$x=\frac{-10±\sqrt{10^{2}-4\times 16}}{2}$$

Square 10.

$$x=\frac{-10±\sqrt{100-4\times 16}}{2}$$

Multiply -4 times 16.

$$x=\frac{-10±\sqrt{100-64}}{2}$$

Add 100 to -64.

$$x=\frac{-10±\sqrt{36}}{2}$$

Take the square root of 36.

$$x=\frac{-10±6}{2}$$

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

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

Divide -4 by 2.

$$x=-2$$

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

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

Divide -16 by 2.

$$x=-8$$

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

$$x^{2}+10x+16=\left(x-\left(-2\right)\right)\left(x-\left(-8\right)\right)$$

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

$$x^{2}+10x+16=\left(x+2\right)\left(x+8\right)$$

Evaluate

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