Question

How do you simplify \[\dfrac{x-2}{{{x}^{2}}+3x-10}\]?

Answer 1

Hint: In this problem, we have to simplify the given fraction \[\dfrac{x-2}{{{x}^{2}}+3x-10}\], to its reduced form. Now we can take the denominator and can reduce it. We can factorize the denominator, so that we can get the factors of the quadratic equation, now the denominator has two factors, in which one of the factors of the quadratic equation can be cancelled with the numerator and the remaining fraction or terms are the simplified form.

Complete step by step answer:
We know that the given fraction to be simplified is,
\[\dfrac{x-2}{{{x}^{2}}+3x-10}\]…….. (1)
Here we can take the denominator, which is the quadratic equation
\[{{x}^{2}}+3x-10\]
Now, we can factorize the above quadratic equation to simplify.
The constant term in the quadratic equation is -10, which can be said as \[5\times -2\]
When we add 5 and - 2 we get 3 which is the coefficient of x.
Therefore, the factors of the quadratic are \[\left( x+5 \right)\left( x-2 \right)\].
These factors can be written in the denominator of (1), instead of the quadratic equation.
We can write (1) as,
\[\Rightarrow \dfrac{x-2}{\left( x-2 \right)\left( x+5 \right)}\]
Now we can cancel the similar terms in the above fraction, we get
\[\Rightarrow \dfrac{1}{x-5}\]

Therefore, the answer is \[\dfrac{1}{x-5}\].

Note: Students make mistakes in finding the factors of the quadratic equation, in this problem it is the perfect square equation, so we can find the factors by normal factorization and can cancel with the similar terms, if it is not a perfect square then we have to find the factor in another method.