Question

What is x^2 + 4x + 4 ——————— x^2 - 5x -14 in simplest form? State any restrictions on the variable. The simplest form is _______.

Answer 1

Answer:$\begin{gathered} Simplest\text{ form: }\frac{x+2}{x-7} \\ \\ Restriction:\text{ x ^^^^2260 7} \end{gathered}$Explanation:

The given expression is:

$\frac{x^2+4x+4}{x^2-5x-14}$

Factorize the numerator

$\begin{gathered} x^2+4x+4 \\ \\ x^2+2x+2x+4 \\ \\ x(x+2)+2(x+2) \\ \\ (x+2)(x+2) \end{gathered}$

Factorize the denominator

$\begin{gathered} x^2-5x-14 \\ \\ x^2-7x+2x-14 \\ \\ x(x-7)+2(x-7) \\ \\ (x+2)(x-7) \end{gathered}$

The given quotient expression then becomes:

$\begin{gathered} \frac{(x+2)(x+2)}{(x+2)(x-7)} \\ \\ \frac{x+2}{x-7} \end{gathered}$

The simplest form is:

$\frac{x+2}{x-7}$

The restriction is:

x ≠ 7