Question

Simplify completely quantity $x$ squared plus $4x$ minus $45$all over $x$ squared plus $10x$ plus $9$ and find the restrictions on the variable.

Answer 1

Step-1: factoring the numerator and denominator:

$\frac{(x^2+4x-45)}{(x^2+10x+9)}$
Proceeding the numerator $x^2+4x-45$
$$\begin{gathered} =(x^2+9x)-(5x+45) \\ =x(x+9)-5(x+9) \\ =(x+9)(x-5) \end{gathered}$$
Proceeding the denominator $x^2+10x+9$
$$\begin{gathered} =x^2+x+9x+9 \\ =(x^2+x)+(9x+9) \\ =x(x+1)+9(x+1) \\ =(x+1)(x+9) \end{gathered}$$

Step-2:Now write a factorized form of the expression:

$$\begin{gathered} \Rightarrow \frac{(x²+4x-45)}{(x²+10x+9)}=\frac{(x+9)(x-5)}{(x+1)(x+9)} \\ =\frac{(x-5)}{(x+1)} \end{gathered}$$

As we can see that as $x=-1,-9$ then the expression becomes undefined.

Hence, simplified form is $\frac{(x-5)}{(x+1)}$and the restriction on the variables are $x\ne -1,-9$.