Question

simplify the expression completely if possible.
$\frac{x^{2}+18x + 81}{2x^{2}+14x}$

Explanation:

Step1: Factor the numerator

The numerator $x^{2}+18x + 81$ is a perfect - square trinomial. Using the formula $(a + b)^2=a^{2}+2ab + b^{2}$, where $a = x$ and $b = 9$, we get $x^{2}+18x + 81=(x + 9)^{2}$.

Step2: Factor the denominator

Factor out the greatest common factor from the denominator $2x^{2}+14x$. The GCF of $2x^{2}$ and $14x$ is $2x$, so $2x^{2}+14x=2x(x + 7)$.

Step3: Simplify the rational expression

The original expression $\frac{x^{2}+18x + 81}{2x^{2}+14x}$ becomes $\frac{(x + 9)^{2}}{2x(x + 7)}$. Since there are no common factors between the numerator and the denominator (other than 1), this is the simplified form.

Answer:

$\frac{(x + 9)^{2}}{2x(x + 7)}$