Question

How do you simplify the expression $\dfrac{{x + 5}}{{{x^2} + 7x + 10}}$?

Answer 1

Hint: Start with factoring the quadratic equation in the denominator by splitting the middle term into two terms. Then cancel out the common factors from numerator and denominator to get the final simplified form of the expression.

Complete step-by-step solution:
According to the question, an algebraic expression is given to us and we have to show the process of its simplification.
Let the given expression be denoted by a variable $y$. So we have:
$ \Rightarrow y = \dfrac{{x + 5}}{{{x^2} + 7x + 10}}{\text{ }}.....{\text{(1)}}$
First we’ll factorize the quadratic equation in the denominator by splitting the middle term into two terms. And the magnitude of the product of the coefficient of those terms must be equal to the magnitude of the product of the coefficients of the first and last terms of the quadratic equation. Keeping this in mind, we’ll factorize it as shown below:
$ \Rightarrow {x^2} + 7x + 10 = {x^2} + 5x + 2x + 10$
Taking common factors from the first two terms and the last two terms, we’ll get:
$
   \Rightarrow {x^2} + 7x + 10 = x\left( {x + 5} \right) + 2\left( {x + 5} \right) \\
   \Rightarrow {x^2} + 7x + 10 = \left( {x + 5} \right)\left( {x + 2} \right)
 $
Now putting the final factored form of the quadratic equation in equation (1), we’ll get:
$ \Rightarrow y = \dfrac{{x + 5}}{{\left( {x + 5} \right)\left( {x + 2} \right)}}$
We can see that $\left( {x + 5} \right)$ is a common factor of both numerator and denominator. So cancelling it out, we’ll get:
$ \Rightarrow y = \dfrac{1}{{\left( {x + 2} \right)}}$
Replacing $y$ with original expression, we have:
$ \Rightarrow \dfrac{{x + 5}}{{{x^2} + 7x + 10}} = \dfrac{1}{{\left( {x + 2} \right)}}$

Thus the simplified form of the given expression is $\dfrac{1}{{\left( {x + 2} \right)}}$.

Note: If we are facing any difficulty in factoring a quadratic equation, we can also use a direct formula to find its roots and then write it in factors. Let the quadratic equation be:
$ \Rightarrow y = a{x^2} + bx + c$
And let $p$ and $q$ are its roots, then the factored form of quadratic equation $y$ can be written as:
$ \Rightarrow a{x^2} + bx + c = a\left( {x - p} \right)\left( {x - q} \right)$
And the formula to determine the roots is:
$ \Rightarrow x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$