Question

How do you simplify the expression $\dfrac{{{x^2} + 4x - 3}}{{{x^3} + 9{x^2} + 8x}}$?

Answer 1

Hint: In this question we have a fraction in which the numerator and denominator are in the form of polynomial equations. We will factorize the numerator and denominator separately and then then re-writes them again as a fraction.

Formula used: $({x_1},{x_2}) = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$
Where $({x_1},{x_2})$ are the roots of the equation and $a,b,c$ are the coefficients of the terms in the quadratic equation

Complete step-by-step solution:
Let’s consider the numerator of the fraction, it is:
${x^2} + 4x - 3$
Now this polynomial is in the form of a quadratic equation, therefore we will factorize it.
Since the middle term cannot be split we will use the quadratic formula to get the roots of the equation.
From the equation, we have:
$a = 1$
$b = 4$
$c = - 3$
On substituting the values in the quadratic formula, we get:
$ \Rightarrow ({x_1},{x_2}) = \dfrac{{ - 4 \pm \sqrt {{4^2} - 4(1)( - 3)} }}{{2(1)}}$
On simplifying the terms in the root and in the denominator, we get:
$ \Rightarrow ({x_1},{x_2}) = \dfrac{{ - 4 \pm \sqrt {28} }}{2}$
Now we know that $\sqrt {28} = 2\sqrt 7 $therefore, on substituting, we get:
$ \Rightarrow ({x_1},{x_2}) = \dfrac{{ - 4 \pm 2\sqrt 7 }}{2}$
Now on splitting the fraction, we get:
$ \Rightarrow ({x_1},{x_2}) = \dfrac{{ - 4}}{2} \pm \dfrac{{2\sqrt 7 }}{2}$
On simplifying, we get:
$ \Rightarrow ({x_1},{x_2}) = - 2 \pm \sqrt 7 $
Therefore, the roots are:
${x_1} = - 2 + \sqrt 7 $and ${x_2} = - 2 - \sqrt 7 $
Therefore, the numerator can be written in the simplified form as:
$ \Rightarrow (x + 2 - \sqrt 7 )(x + 2 + \sqrt 7 )$.
Now let’s consider the denominator of the fraction:
$ \Rightarrow {x^3} + 9{x^2} + 8x$
Now $x$ is common in all the terms therefore, we can remove it out as common as:
$ \Rightarrow x({x^2} + 9x + 8)$
Now the expression is in the form of a quadratic equation, therefore we will find its roots using the quadratic formula
$a = 1$
$b = 9$
$c = 8$
On substituting the values in the quadratic formula, we get:
$ \Rightarrow ({x_1},{x_2}) = \dfrac{{ - 9 \pm \sqrt {{9^2} - 4(1)(8)} }}{{2(1)}}$
On simplifying the terms in the root and in the denominator, we get:
$ \Rightarrow ({x_1},{x_2}) = \dfrac{{ - 9 \pm \sqrt {49} }}{2}$
Now we know that $\sqrt {49} = 7$ therefore, on substituting, we get:
$ \Rightarrow ({x_1},{x_2}) = \dfrac{{ - 9 \pm 7}}{2}$
Therefore, the roots are:
${x_1} = - 1$ and ${x_2} = - 8$
Therefore, the numerator can be written in the simplified form as:
$ \Rightarrow (x + 1)(x + 8)$.

Therefore, the fraction can be written as:
$ \Rightarrow \dfrac{{(x + 2 - \sqrt 7 )(x + 2 + \sqrt 7 )}}{{(x + 1)(x + 8)}}$, which is the required solution.

Note: It is to be remembered that the quadratic formula should be used when the quadratic equation cannot be solved directly by splitting the middle term.
A quadratic equation is a polynomial equation with a degree $2$, quadratic equations are used mostly in statistics when there is a power.