Chapter 6.6, Problem 2MRE

To determine

Simplify:the given expression.

Answer to Problem 2MRE

The simplified form of $\frac{a^2-9a+14}{49-a^2}$ is $-\frac{\left(a-2\right)}{\left(7+a\right)}$.

Explanation of Solution

Given information:

The expression $\frac{a^2-9a+14}{49-a^2}$.

Formula used:

  $a^2-b^2=\left(a-b\right)\left(a+b\right)$

Splitting the middle term: In a quadratic equation, the middle term of equation is splitted in such a way that product of both component is equal to product of constant and coefficient of quadratic term.

  $\frac{a^2-9a+14}{49-a^2}$

Simplifying numerator by using splitting the middle term method.

Recall that in splitting the middle term method the middle term of equation is splitted in such a way that product of both component is equal to product of constant and coefficient of quadratic term.

  $\begin{array}{l}=\frac{a^2-7a-2a+14}{49-a^2}\\ =\frac{a\left(a-7\right)-2\left(a-7\right)}{49-a^2}\\ =\frac{\left(a-2\right)\left(a-7\right)}{49-a^2}\end{array}$

Factorizing the denominator by using identity $a^2-b^2=\left(a-b\right)\left(a+b\right)$.

  $\begin{array}{l}=\frac{\left(a-2\right)\left(a-7\right)}{7^2-a^2}\\ =\frac{\left(a-2\right)\left(a-7\right)}{\left(7-a\right)\left(7+a\right)}\\ =\frac{\left(a-2\right)\left(a-7\right)}{-\left(a-7\right)\left(7+a\right)}\end{array}$

As $\left(a-7\right)$ is common in numerator and denominator, So, $\left(a-7\right)$ will be cancelled out.

  $=\boxed{-\frac{\left(a-2\right)}{\left(7+a\right)}}$

The simplified form of $\frac{a^2-9a+14}{49-a^2}$ is $-\frac{\left(a-2\right)}{\left(7+a\right)}$.