Question

How do you simplify \[\dfrac{{{x}^{2}}+2x}{5x+10}\]?

Answer 1

Hint: We have one quadratic equation and one linear equation in the numerator and the denominator respectively. We factor them by taking common terms from them. We can take x outside from the numerator and 5 from the denominator. We will then eliminate the common factors. We will hence be able to find the simplified form of the expression.

Complete step by step solution:
We have been given the division of one quadratic equation and one linear equation. We need to find the factor of the equations.
The two quadratic equations are \[{{x}^{2}}+2x\] and \[5x+10\].
 We take commons of variables or constants to factor them.
We take $f\left( x \right)={{x}^{2}}+2x$. We can take $x$ as the common number from the terms.
We get \[{{x}^{2}}+2x=x\left( x+2 \right)\].
For $g\left( x \right)=5x+10$, we can take 5 as the common number from the terms.
We get \[5x+10=5\left( x+2 \right)\].
Now we put the factored values for the two equations.
We get \[\dfrac{{{x}^{2}}+2x}{5x+10}=\dfrac{x\left( x+2 \right)}{5\left( x+2 \right)}\].
Now we can see in the denominator and the numerator the common factor is $\left( x+2 \right)$.
We can eliminate the factor from both denominator and the numerator.
So, \[\dfrac{{{x}^{2}}+2x}{5x+10}=\dfrac{x\left( x+2 \right)}{5\left( x+2 \right)}=\dfrac{x}{5}\]. The only condition being $x\ne -2$.

Therefore, the simplified form of \[\dfrac{{{x}^{2}}+2x}{5x+10}\] is \[\dfrac{x}{5}\].

Note: We find the value of $x=a$ for which the function $f\left( x \right)={{x}^{2}}+2x=0$.
We take $x=a=-2$. We can see $f\left( -2 \right)={{\left( -2 \right)}^{2}}+2\left( -2 \right)=4-4=0$.
So, the root of the $f\left( x \right)={{x}^{2}}+2x$ will be the function $\left( x+2 \right)$. This means for $x=a$, if $f\left( a \right)=0$ then $\left( x-a \right)$ is a root of $f\left( x \right)$.
Therefore, the term $\left( x+2 \right)$ is a factor of the polynomial \[{{x}^{2}}+2x\].
So, \[{{x}^{2}}+2x=x\left( x+2 \right)\].