Question

If we have an equation as \[\ a + \dfrac{1}{a} = 6\] and \[\ a \neq 0\] then find \[a^{2} - \dfrac{1}{a^{2}}\]
A). \[\pm 24\sqrt{3}\]
B). \[\pm 4\sqrt{2}\]
C). \[\pm 24\sqrt{2}\]
D). \[\pm 4\sqrt{3}\]

Answer 1

Hint: Given , \[\ a + \dfrac{1}{a} = 6\] and also given \[a \neq 0\] .First we need to square the given equation . By squaring the equation on both sides we will require a value. Then we need to substitute the value in and then by simplifying we require the answer.
Formula used :
\[\left( a + b \right)^{2} = a^{2} + b^{2} + 2ab\]
\[\left( a – b \right)^{2} = a^{2} + b^{2} – 2ab\]
\[a^{2} – b^{2} = \left( a + b \right)\left( a – b \right)\]

Complete step-by-step solution:
Given, \[ a + \dfrac{1}{a} = 6\]
By squaring on both sides,
We get,
\[\left( a + \dfrac{1}{a} \right)^{2} = 6^{2}\]
Expanding the equation,
We get,
\[\left( a^{2} + \left( \dfrac{1}{a} \right)^{2} + 2a\left( \dfrac{1}{a} \right) \right) = 36\]
By simplifying,
We get,
\[a^{2} + \dfrac{1}{a^{2}} + 2 = 36\]
\[a^{2} + \dfrac{1}{a^{2}} = 36 – 2\]
By subtracting,
We get,
\[a^{2} + \dfrac{1}{a^{2}} = 34\]
Now we have to find the value of \[a^{2} - \dfrac{1}{a^{2}},\]
Now we can consider \[\left( a - \dfrac{1}{a} \right)^{2}\]
By expanding,
We get,
\[\left( a - \dfrac{1}{a} \right)^{2} = a^{2} – 2\left( a \right)\left( \dfrac{1}{a} \right) + \left( \dfrac{1}{a} \right)^{2}\]
By simplifying,
We get,
\[\left( a - \dfrac{1}{a} \right)^{2} = a^{2} – 2 + \dfrac{1}{a^{2}}\]
By rearranging,
We get,
\[\left( a - \dfrac{1}{a} \right)^{2} = a^{2} + \dfrac{1}{a^{2}} – 2\]
We have already found the value of \[a^{2} + \dfrac{1}{a^{2}} = 34\]
Now by substituting the value,
We get,
\[\left( a - \dfrac{1}{a} \right)^{2} = 34 – 2\]
By simplifying,
We get,
\[\left( a - \dfrac{1}{a} \right)^{2} = 32\]
By taking square root on both sides,
We get,
\[\sqrt{\left( a - \dfrac{1}{a} \right)^{2}} = \pm \sqrt{32}\]
By simplifying,
We get,
\[\left( a - \dfrac{1}{a} \right) = \pm 4\sqrt{2}\]
Now , we need to find \[\left( a^{2} - \dfrac{1}{a^{2}} \right)\]
W. K. T: \[a^{2} – b^{2} = \left( a + b \right)\left( a – b \right)\]
Thus
\[a^{2} - \dfrac{1}{a^{2}} = \left( a + \dfrac{1}{a} \right)\left( a - \dfrac{1}{a} \right)\]
Now we can substitute the known values.
\[a^{2} - \dfrac{1}{a^{2}} = 6\left( \pm 4\sqrt{2} \right)\]
By multiplying,
We get,
\[a^{2} - \dfrac{1}{a^{2}} = \pm 24\sqrt{2}\]

The value of \[a^{2} - \dfrac{1}{a^{2}} = \pm 24\sqrt{2}\]
Option (C). \[{\pm}24\sqrt{2}\] is correct.

Note: The concept used to solve this problem is algebraic Identities and substitution method. Algebraic Identities are used for the Factorization of the polynomials. There are eight more algebraic identities used for the factorization of the polynomial.