Solve x^2+12x+27

$${ x }^{ 2 } +12x+27$$

Factor

$$\left(x+3\right)\left(x+9\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx+27. To find a and b, set up a system to be solved.

$$a+b=12 ab=1\times 27=27$$

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 27.

$$1,27 3,9$$

Calculate the sum for each pair.

$$1+27=28 3+9=12$$

The solution is the pair that gives sum 12.

$$a=3 b=9$$

Rewrite x^{2}+12x+27 as \left(x^{2}+3x\right)+\left(9x+27\right).

$$\left(x^{2}+3x\right)+\left(9x+27\right)$$

Factor out x in the first and 9 in the second group.

$$x\left(x+3\right)+9\left(x+3\right)$$

Factor out common term x+3 by using distributive property.

$$\left(x+3\right)\left(x+9\right)$$

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.

$$x^{2}+12x+27=0$$

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.

$$x=\frac{-12±\sqrt{12^{2}-4\times 27}}{2}$$

Square 12.

$$x=\frac{-12±\sqrt{144-4\times 27}}{2}$$

Multiply -4 times 27.

$$x=\frac{-12±\sqrt{144-108}}{2}$$

Add 144 to -108.

$$x=\frac{-12±\sqrt{36}}{2}$$

Take the square root of 36.

$$x=\frac{-12±6}{2}$$

Now solve the equation x=\frac{-12±6}{2} when ± is plus. Add -12 to 6.

$$x=-\frac{6}{2}$$

Divide -6 by 2.

$$x=-3$$

Now solve the equation x=\frac{-12±6}{2} when ± is minus. Subtract 6 from -12.

$$x=-\frac{18}{2}$$

Divide -18 by 2.

$$x=-9$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -3 for x_{1} and -9 for x_{2}.

$$x^{2}+12x+27=\left(x-\left(-3\right)\right)\left(x-\left(-9\right)\right)$$

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

$$x^{2}+12x+27=\left(x+3\right)\left(x+9\right)$$

Evaluate

$$\left(x+3\right)\left(x+9\right)$$