Question

Simplify $\dfrac{{\dfrac{x}{4} - \dfrac{1}{x}}}{{\dfrac{1}{{2x}} + \dfrac{1}{4}}}$ ?

Answer 1

Hint: Simplify the given expression and this can be done by taking L.C.M in both the numerator and denominator, and by eliminating the like terms we will get an expression and then we will use the algebraic identity ${a^2} - {b^2} = \left( {a - b} \right)\left( {a + b} \right)$, then by simplifying further we will get the required result.

Complete step by step answer:
Rational expression is the ratio of two polynomials. In rational expression both the numerator and denominator are polynomials. The denominator of rational expression can never be zero.
Given expression is $\dfrac{{\dfrac{x}{4} - \dfrac{1}{x}}}{{\dfrac{1}{{2x}} + \dfrac{1}{4}}}$,
Now taking least common factor i.e., L.C.M n both numerator and denominator to make the denominators of the numerators and denominators equal we get,
$ \Rightarrow \dfrac{{\dfrac{{x \times x}}{{4 \times x}} - \dfrac{{1 \times 4}}{{x \times 4}}}}{{\dfrac{{1 \times 4}}{{2x \times 4}} + \dfrac{{1 \times 2x}}{{4 \times 2x}}}}$,
Now simplifying we get,
$ \Rightarrow \dfrac{{\dfrac{{{x^2}}}{{4x}} - \dfrac{4}{{4x}}}}{{\dfrac{4}{{8x}} + \dfrac{{2x}}{{8x}}}}$,
Again simplifying we get,
$ \Rightarrow \dfrac{{\dfrac{{{x^2} - 4}}{{4x}}}}{{\dfrac{{2x + 4}}{{8x}}}}$,
Now taking the like terms in common in the denominator, we get,
$ \Rightarrow \dfrac{{\dfrac{{{x^2} - 4}}{{4x}}}}{{\dfrac{{2\left( {x + 2} \right)}}{{8x}}}}$,
Now simplifying we get,
$ \Rightarrow \dfrac{{\dfrac{{{x^2} - 4}}{{4x}}}}{{\dfrac{{x + 2}}{{4x}}}}$,
Now eliminating the like terms as both the numerator and denominator has same denominator we get,
$ \Rightarrow \dfrac{{{x^2} - 4}}{{x + 2}}$,
Now rewriting the numerator we get,
$ \Rightarrow \dfrac{{\left( {{x^2}} \right) - {{\left( 2 \right)}^2}}}{{x + 2}}$,
Now using the identity ${a^2} - {b^2} = \left( {a - b} \right)\left( {a + b} \right)$ in the numerator we get,
Here $a = x$ and $b = 2$, we get,
$ \Rightarrow \dfrac{{\left( {x - 2} \right)\left( {x + 2} \right)}}{{x + 2}}$,
Now eliminating the like terms we get,
\[ \Rightarrow x - 2\],
So, the simplified form of the given expression is \[x - 2\].

\[\therefore \] The simplified form of the given expression $\dfrac{{\dfrac{x}{4} - \dfrac{1}{x}}}{{\dfrac{1}{{2x}} + \dfrac{1}{4}}}$ will be equal to \[x - 2\].

Note: To simplify the rational expression we reduce it to its simplest form. The quotient of two polynomials expressions is called a rational expression. Simplifying the rational expression means to reduce to its lowest terms. Rational expressions are in their lowest form if all common factors from the numerator and denominator have been cancelled. Solving the rational expressions is more complex than solving standard polynomial equations because we have to find the common denominator of the rational terms, then simplify the resulting expressions.