Question

How do you simplify the expression $\dfrac{15{{x}^{2}}-8x-18}{-20{{x}^{2}}+14x+12}$

Answer 1

Hint: Now to simplify the expression we will factorize the quadratic in numerator and denominator. Now we know that the roots of the expression of the form $a{{x}^{2}}+bx+c$ is given by the formula $\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$ . Hence using this formula we will find the roots of the expression. Now we know that $\alpha $ and $\beta $ are the roots of the expression then $x-\alpha $ and $x-\beta $ are the factors of the expression. Hence we can find the factors of the expression and rewrite the fraction as factors obtained.

Complete step by step answer:
Now first let us consider the numerator.
The numerator is $15{{x}^{2}}-8x-18$ .
This is a quadratic expression in x of the form $a{{x}^{2}}+bx+c$ .
Now we know that the roots of the quadratic expressions are given by $\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$
Hence the roots of the given expression are
$\begin{align}
  & \Rightarrow x=\dfrac{8\pm \sqrt{{{\left( -8 \right)}^{2}}-4\left( 15 \right)\left( -18 \right)}}{2\left( 15 \right)} \\
 & \Rightarrow x=\dfrac{8\pm \sqrt{64+1080}}{30} \\
 & \Rightarrow x=\dfrac{8\pm 2\sqrt{286}}{30} \\
 & \Rightarrow x=\dfrac{4\pm \sqrt{286}}{15} \\
\end{align}$
Hence the roots of the expression are $x=\dfrac{4+\sqrt{286}}{15}$ and $x=\dfrac{4-\sqrt{286}}{15}$ .
Now hence we get the factors of the expression as $\left( x-\dfrac{4+\sqrt{286}}{15} \right)$ and $\left( x+\dfrac{4+\sqrt{286}}{15} \right)$
Now consider the expression in denominator $-20{{x}^{2}}+14x+12$
Now let us try to find the factors of the expression.
We know that again the given expression is a quadratic expression of the form $a{{x}^{2}}+bx+c$ whose roots are given by $\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$ .
Hence we can say that the roots of the above expression are,
$\begin{align}
  & \Rightarrow x=\dfrac{-14\pm \sqrt{{{14}^{2}}-4\left( -20 \right)\left( 12 \right)}}{2\left( -20 \right)} \\
 & \Rightarrow x=\dfrac{-14\pm \sqrt{196+960}}{-40} \\
 & \Rightarrow x=\dfrac{-14\pm \sqrt{1156}}{-40} \\
 & \Rightarrow x=\dfrac{-14\pm 34}{40} \\
 & \Rightarrow x=\dfrac{-14-34}{40},x=\dfrac{-14+34}{40} \\
 & \Rightarrow x=-10,x=\dfrac{1}{2} \\
\end{align}$
Hence the factors of the given expression are $\left( x+10 \right)$ and $\left( x-\dfrac{1}{2} \right)$
Now rewriting the expression as their factors we get,
$\dfrac{\left( x-\dfrac{4+\sqrt{286}}{15} \right)\left( x+\dfrac{4+\sqrt{286}}{15} \right)}{\left( x+10 \right)\left( x-\dfrac{1}{2} \right)}$
Hence the given expression is simplified.

Note: Now note that we can also find the roots of the equation by using the complete square method. In this method we will add and subtract ${{\left( \dfrac{b}{2a} \right)}^{2}}$ on both sides of the monic expression and then simplify the expression to get a complete square. Then we further simplify by taking square root and hence find the value of x.