Solve b^2+12b+32

$${ b }^{ 2 } +12b+32$$

Factor

$$\left(b+4\right)\left(b+8\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as b^{2}+pb+qb+32. To find p and q, set up a system to be solved.

$$p+q=12 pq=1\times 32=32$$

Since pq is positive, p and q have the same sign. Since p+q is positive, p and q are both positive. List all such integer pairs that give product 32.

$$1,32 2,16 4,8$$

Calculate the sum for each pair.

$$1+32=33 2+16=18 4+8=12$$

The solution is the pair that gives sum 12.

$$p=4 q=8$$

Rewrite b^{2}+12b+32 as \left(b^{2}+4b\right)+\left(8b+32\right).

$$\left(b^{2}+4b\right)+\left(8b+32\right)$$

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

$$b\left(b+4\right)+8\left(b+4\right)$$

Factor out common term b+4 by using distributive property.

$$\left(b+4\right)\left(b+8\right)$$

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.

$$b^{2}+12b+32=0$$

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.

$$b=\frac{-12±\sqrt{12^{2}-4\times 32}}{2}$$

Square 12.

$$b=\frac{-12±\sqrt{144-4\times 32}}{2}$$

Multiply -4 times 32.

$$b=\frac{-12±\sqrt{144-128}}{2}$$

Add 144 to -128.

$$b=\frac{-12±\sqrt{16}}{2}$$

Take the square root of 16.

$$b=\frac{-12±4}{2}$$

Now solve the equation b=\frac{-12±4}{2} when ± is plus. Add -12 to 4.

$$b=-\frac{8}{2}$$

Divide -8 by 2.

$$b=-4$$

Now solve the equation b=\frac{-12±4}{2} when ± is minus. Subtract 4 from -12.

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

Divide -16 by 2.

$$b=-8$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -4 for x_{1} and -8 for x_{2}.

$$b^{2}+12b+32=\left(b-\left(-4\right)\right)\left(b-\left(-8\right)\right)$$

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

$$b^{2}+12b+32=\left(b+4\right)\left(b+8\right)$$

Evaluate

$$\left(b+4\right)\left(b+8\right)$$