Question

If ${a^2} + \dfrac{1}{{{a^2}}} = 14$, find ${a^2} - \dfrac{1}{{{a^2}}}.$

Answer 1

Hint: Here, using two different formula for the same given mathematical statement, find the values of $\left( {a + \dfrac{1}{a}} \right)$ and $\left( {a - \dfrac{1}{a}} \right)$ and multiply them to get the result.

Complete step-by-step answer:
Given that ${a^2} + \dfrac{1}{{{a^2}}} = 14$
$ \Rightarrow {\left( {a + \dfrac{1}{a}} \right)^2} - 2 \times a \times \dfrac{1}{a} = 14$ [Since, ${a^2} + {b^2} = {(a + b)^2} - 2ab$]
On simplifying
$ \Rightarrow {\left( {a + \dfrac{1}{a}} \right)^2} - 2 = 14$
Adding 2 to both sides
$ \Rightarrow {\left( {a + \dfrac{1}{a}} \right)^2} = 16$
Taking square root on both sides, we get
$\left( {a + \dfrac{1}{a}} \right) = \pm 4$ …(i)
Now, different formula to find the value of $\left( {a - \dfrac{1}{a}} \right)$
${a^2} + \dfrac{1}{{{a^2}}} = 14$
$ \Rightarrow {\left( {a - \dfrac{1}{a}} \right)^2} + 2 \times a \times \dfrac{1}{a} = 14$ [Since, ${a^2} + {b^2} = {(a - b)^2} + 2ab$]
On simplifying
$ \Rightarrow {\left( {a - \dfrac{1}{a}} \right)^2} + 2 = 14$
Subtracting 2 to both sides
$ \Rightarrow {\left( {a - \dfrac{1}{a}} \right)^2} = 12$
Taking square root on both sides, we get
$\left( {a - \dfrac{1}{a}} \right) = \pm 2\sqrt 3 $ …(ii)
Multiplying equations (i) and (ii), we get
$\left( {a - \dfrac{1}{a}} \right)\left( {a + \dfrac{1}{a}} \right) = \pm 4 \times \pm 2\sqrt 3 = \pm 8\sqrt 3 $
$ \Rightarrow {a^2} - \dfrac{1}{{{a^2}}} = \pm 8\sqrt 3 $

Therefore, required value is $ \pm 8\sqrt 3 $

Note: In these types of questions, choose the formula very carefully. Simplify the terms asked to find and relate the values from the value given in question. Do not try to find the value of a variable until not asked to find the value, only focus on what is asked and relate. Also careful about signs, always remember whenever we do square root operation, both + and – sign will be considered.
Alternatively, we can find the value of a variable and simply put the values in the equation to get the result. Here in this question equation (i) is
$\left( {a + \dfrac{1}{a}} \right) = \pm 4$
Form the above as quadratic equation
${a^2} + 1 = \pm 4a$
Now, we have two equations
${a^2} - 4a + 1 = 0$ and ${a^2} + 4a + 1 = 0$
By solving these two equations we get the value of a, then put the values of a in the equation to get the result. This method is slightly long but easy to do. If two variables are given then you may have to find the values of both variables. Use trick and hit and trial methods to solve these types of questions.