Question

If we have an equation as \[x - \dfrac{1}{x} = 8\], then the value of \[{x^2} + \dfrac{1}{{{x^2}}}\] is :
A). \[10\]
B). \[62\]
C). \[6\]
D). \[66\]

Answer 1

Hint: First we have to define what the terms we need to solve the problem are. Here we use the formula of \[{(a - b)^2}\] which is \[{a^2} + {b^2} - 2ab\] and use the transposition of the term to get the needed terms.

Complete step-by-step solution:
There are two ways to find the value of \[{x^2} + \dfrac{1}{{{x^2}}}\] with the help of given equation that is \[x - \dfrac{1}{x} = 8\].
One way is to first find the value of \[x\] with the help of given equation \[x - \dfrac{1}{x} = 8\],and once it is found we will put the value of \[x\] in the problem equation to find out its value, another way if to expand the given equation by squaring both side and simplifying. Here we are going to see the second method.
Since it is given that
\[x - \dfrac{1}{x} = 8\]
if we take \[x\]in place of \[a\] and \[\dfrac{1}{x}\] in place of b we can get \[{x^2} + \dfrac{1}{{{x^2}}}\] simply by the whole square formula provided above.
So, As per the above formula if we expand \[x - \dfrac{1}{x}\] we can get the value of \[{x^2} + \dfrac{1}{{{x^2}}}\]
Since we have the value of \[x - \dfrac{1}{x}\] , by putting its value on the equation we can get the value of\[{x^2} + \dfrac{1}{{{x^2}}}\]
So,
\[\Rightarrow {(x - \dfrac{1}{x})^2} = {x^2} + \dfrac{1}{{{x^2}}} - 2.x.\dfrac{1}{x}\]
xpanding equation according to the above formula we get
\[\Rightarrow {(x - \dfrac{1}{x})^2} = {x^2} + \dfrac{1}{{{x^2}}} - 2\]
simplifying the equation, we get
\[\Rightarrow {8^2} = {x^2} + \dfrac{1}{{{x^2}}} - 2\] \[\because \]\[x - \dfrac{1}{x}\]\[ = 8\]
putting its value in the equation we get
\[\Rightarrow 64 + 2 = {x^2} + \dfrac{1}{{{x^2}}}\]
Simplifying the equation, adding constant terms we get
\[\Rightarrow {x^2} + \dfrac{1}{{{x^2}}} = 66\]
Hence, the correct option is (D) \[66\].

Note: When expanding the equation, simplify it in each step if possible. There is one more method through which we can find the value of \[{x^2} + \dfrac{1}{{{x^2}}}\]. Since we know the value of \[x - \dfrac{1}{x} = 8\] we will simply find the value of \[x\] and put it in \[{x^2} + \dfrac{1}{{{x^2}}}\] to get its value.