Solve x^2+16x+15

$${ x }^{ 2 } +16x+15$$

Factor

$$\left(x+1\right)\left(x+15\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx+15. To find a and b, set up a system to be solved.

$$a+b=16 ab=1\times 15=15$$

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 15.

$$1,15 3,5$$

Calculate the sum for each pair.

$$1+15=16 3+5=8$$

The solution is the pair that gives sum 16.

$$a=1 b=15$$

Rewrite x^{2}+16x+15 as \left(x^{2}+x\right)+\left(15x+15\right).

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

Factor out x in the first and 15 in the second group.

$$x\left(x+1\right)+15\left(x+1\right)$$

Factor out common term x+1 by using distributive property.

$$\left(x+1\right)\left(x+15\right)$$

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.

$$x^{2}+16x+15=0$$

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.

$$x=\frac{-16±\sqrt{16^{2}-4\times 15}}{2}$$

Square 16.

$$x=\frac{-16±\sqrt{256-4\times 15}}{2}$$

Multiply -4 times 15.

$$x=\frac{-16±\sqrt{256-60}}{2}$$

Add 256 to -60.

$$x=\frac{-16±\sqrt{196}}{2}$$

Take the square root of 196.

$$x=\frac{-16±14}{2}$$

Now solve the equation x=\frac{-16±14}{2} when ± is plus. Add -16 to 14.

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

Divide -2 by 2.

$$x=-1$$

Now solve the equation x=\frac{-16±14}{2} when ± is minus. Subtract 14 from -16.

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

Divide -30 by 2.

$$x=-15$$

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

$$x^{2}+16x+15=\left(x-\left(-1\right)\right)\left(x-\left(-15\right)\right)$$

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

$$x^{2}+16x+15=\left(x+1\right)\left(x+15\right)$$

Evaluate

$$\left(x+1\right)\left(x+15\right)$$