Question

How do you simplify $\dfrac{{{x^2} - 16}}{{{x^2} - 4x}}$?

Answer 1

Hint: We will first find the factors of numerator of the given function using the identity given by ${a^2} - {b^2} = \left( {a - b} \right)\left( {a + b} \right)$. Then, we will find factors of the denominator by taking out x common and thus, we have a simplified answer.

Complete step by step solution:
We are given that we are required to simplify $\dfrac{{{x^2} - 16}}{{{x^2} - 4x}}$.
Let us assume that $f(x) = \dfrac{{{x^2} - 16}}{{{x^2} - 4x}}$.
The numerator of the function f (x) is given by the expression ${x^2} - 16$.
Since, we know that we have an identity given by the following expression with us:-
$ \Rightarrow {a^2} - {b^2} = \left( {a - b} \right)\left( {a + b} \right)$
Replacing a by $x$ and b by 4 in the above mentioned equation, we will then obtain the following equation with us:-
$ \Rightarrow {x^2} - {4^2} = \left( {x - 4} \right)\left( {x + 4} \right)$
Simplifying the left hand side of the above equation, we will then obtain the following equation with us:-
$ \Rightarrow {x^2} - 16 = \left( {x - 4} \right)\left( {x + 4} \right)$
Replacing this in the function f (x), we have the following equation with us:-
$ \Rightarrow f(x) = \dfrac{{(x - 4)(x + 4)}}{{{x^2} - 4x}}$
The denominator of the function f (x) is given by ${x^2} - 4x$.
Taking x common from it, we will then obtain the following equation with us:-
$ \Rightarrow {x^2} - 4x = x(x - 4)$
Putting this in f (x) as well, we will then obtain the following equation with us:-
$ \Rightarrow f(x) = \dfrac{{(x - 4)(x + 4)}}{{x(x - 4)}}$
Crossing – off (x – 4) from both the numerator and denominator, we will then obtain the following equation with us:-
$ \Rightarrow f(x) = \dfrac{{x + 4}}{x}$
Simplifying the right hand side further, we will then obtain the following equation with us:-

$ \Rightarrow f(x) = 1 + \dfrac{4}{x}$

Thus, we have the required simplification.

Note:
The students must commit to memory the following identity used in the above solution:-
${a^2} - {b^2} = \left( {a - b} \right)\left( {a + b} \right)$
The students must also notice that we could cross – off the factor (x – 4) from the given function only because we know that x can never be equal to 4, because if x = 4 was possible then the function would not have been defined.
In the simplification in the last steps, we used the fact that: $\dfrac{{a + b}}{c} = \dfrac{a}{c} + \dfrac{b}{c}$.