Question

If \[{\text{x = 5 - 2}}\sqrt {\text{6}} \]. Find the value of: \[{\left( {{\text{x - }}\dfrac{{\text{1}}}{{\text{x}}}} \right)^{\text{2}}}\]

Answer 1

Hint: Here, in this question we are going to find the value of the required data
First, we are using the value of \[{\text{x}}\] and substituting them into the required relation.
And you need to simplify the value that you get,
Also, we need to use some algebraic identities there, to solve them
Then we will get the result which we required.

Formula used: The Formulas which we are using to find the required answer are,
\[{{\text{(a - b)}}^{\text{2}}}{\text{ = }}{{\text{a}}^{\text{2}}}{\text{ + }}{{\text{b}}^{\text{2}}}{\text{ - 2ab}}\]
\[{{\text{(a + b)}}^{\text{2}}}{\text{ = }}{{\text{a}}^{\text{2}}}{\text{ + }}{{\text{b}}^{\text{2}}}{\text{ + 2ab}}\]

Complete step-by-step answer:
In the question they have given that
\[{\text{x = 5 - 2}}\sqrt {\text{6}} \]
Also,
$\Rightarrow$\[\dfrac{{\text{1}}}{{\text{x}}}{\text{ = }}\dfrac{{\text{1}}}{{{\text{5 - 2}}\sqrt {\text{6}} }}\]
By taking conjugate,
$\Rightarrow$\[\dfrac{{\text{1}}}{{\text{x}}}{\text{ = }}\dfrac{{\text{1}}}{{{\text{5 - 2}}\sqrt {\text{6}} }}{{ \times }}\dfrac{{{\text{5 + 2}}\sqrt {\text{6}} }}{{{\text{5 + 2}}\sqrt {\text{6}} }}\]
Here, we are simplifying the value on above we will get
$\Rightarrow$\[\dfrac{{\text{1}}}{{\text{x}}}{\text{ = }}\dfrac{{{\text{5 + 2}}\sqrt {\text{6}} }}{{{{{\text{(5)}}}^{\text{2}}}{\text{ - (2}}\sqrt {\text{6}} {{\text{)}}^{\text{2}}}}}\]
\[\dfrac{{\text{1}}}{{\text{x}}}{\text{ = }}\dfrac{{{\text{5 + 2}}\sqrt {\text{6}} }}{{{\text{25 - 24}}}}\]
By Subtracting we have
$\Rightarrow$\[\dfrac{{\text{1}}}{{\text{x}}}{\text{ = }}\dfrac{{{\text{5 + 2}}\sqrt {\text{6}} }}{1}\]
As, we know any number divided by \[{\text{1}}\], we will get same number
$\Rightarrow$\[\dfrac{{\text{1}}}{{\text{x}}}{\text{ = 5 + 2}}\sqrt {\text{6}} \]
Now we have all the values we are going to substitute them,
$\Rightarrow$\[{\left( {{\text{x - }}\dfrac{1}{{\text{x}}}} \right)^2}\]\[{\text{ = }}{\left( {{\text{(5 - 2}}\sqrt {\text{6}} {\text{) - (5 + 2}}\sqrt {\text{6}} {\text{)}}} \right)^{\text{2}}}\]
While seeing this we can note a thing that the above value is in the form of a formula $\Rightarrow$\[{{\text{(a - b)}}^{\text{2}}}{\text{ = }}{{\text{a}}^{\text{2}}}{\text{ + }}{{\text{b}}^{\text{2}}}{\text{ - 2ab}}\]
So, Assuming that
$\Rightarrow$\[{\text{a = (5 - 2}}\sqrt {\text{6}} {\text{)}}\]
$\Rightarrow$\[{\text{b = (5 + 2}}\sqrt {\text{6}} {\text{)}}\]
And applying them on the equation we will have
\[{\text{ = }}\] \[{{\text{(5 - 2}}\sqrt {\text{6}} {\text{)}}^{\text{2}}}{\text{ + (5 + 2}}\sqrt {\text{6}} {{\text{)}}^{\text{2}}}{\text{ - 2 (5 - 2}}\sqrt {\text{6}} {\text{)(5 + 2}}\sqrt {\text{6}} {\text{)}}\]
While Simplifying them we will get
\[{\text{ = }}\] \[25 + 24 - 2 \times 5 \times 2\sqrt 6 + 25 + 24 + 2 \times 5 \times 2\sqrt 6 - 2(25 + 10\sqrt 6 - 10\sqrt 6 - 24)\]
\[{\text{ = }}\] \[25 + 24 - 2 \times 5 \times 2\sqrt 6 + 25 + 24 + 2 \times 5 \times 2\sqrt 6 - 50 - 20\sqrt 6 + 20\sqrt 6 + 48\]
By Subtraction and multiplication, we have
\[{\text{ = }}\]\[24 + 24 + 48\]
By adding the above values, we get,
\[{\text{ = 96}}\]
The value of the given term is \[96\]

\[\therefore {\left( {{\text{x - }}\dfrac{{\text{1}}}{{\text{x}}}} \right)^{\text{2}}} = 96\]

Note: We are having an alternate method for this problem
\[{\text{x = 5 - 2}}\sqrt {\text{6}} \]
\[\dfrac{{\text{1}}}{{\text{x}}}{\text{ = }}\dfrac{{\text{1}}}{{{\text{5 - 2}}\sqrt {\text{6}} }}\]
By taking conjugate,
$\Rightarrow$\[\dfrac{{\text{1}}}{{\text{x}}}{\text{ = }}\dfrac{{\text{1}}}{{{\text{5 - 2}}\sqrt {\text{6}} }}{{ \times }}\dfrac{{{\text{5 + 2}}\sqrt {\text{6}} }}{{{\text{5 + 2}}\sqrt {\text{6}} }}\]
Here, we are simplifying the value on above we will get
$\Rightarrow$\[\dfrac{{\text{1}}}{{\text{x}}}{\text{ = }}\dfrac{{{\text{5 + 2}}\sqrt {\text{6}} }}{{{{{\text{(5)}}}^{\text{2}}}{\text{ - (2}}\sqrt {\text{6}} {{\text{)}}^{\text{2}}}}}\]
\[\dfrac{{\text{1}}}{{\text{x}}}{\text{ = }}\dfrac{{{\text{5 + 2}}\sqrt {\text{6}} }}{{{\text{25 - 24}}}}\]
By Subtracting we have
$\Rightarrow$\[\dfrac{{\text{1}}}{{\text{x}}}{\text{ = }}\dfrac{{{\text{5 + 2}}\sqrt {\text{6}} }}{1}\]
As, we know any number divided by \[{\text{1}}\], we will get same number
$\Rightarrow$\[\dfrac{1}{{\text{x}}}{\text{ = 5 + 2}}\sqrt {\text{6}} \]
Here we are subtracting the both values we get,
$\Rightarrow$\[\left( {{\text{x - }}\dfrac{1}{{\text{x}}}} \right)\] \[{\text{ = (5 - 2}}\sqrt {\text{6}} {\text{) - (5 + 2}}\sqrt {\text{6}} {\text{)}}\]
By Simplifying them, we get
\[{\text{ = 5 - 2}}\sqrt {\text{6}} {\text{ - 5 - 2}}\sqrt {\text{6}} \]
Here we are subtracting the above values,
$\Rightarrow$\[\left( {{\text{x - }}\dfrac{1}{{\text{x}}}} \right)\] \[{\text{ = - 4}}\sqrt {\text{6}} \]
By squaring on both sides, we have
$\Rightarrow$\[{\left( {{\text{x - }}\dfrac{1}{{\text{x}}}} \right)^2}\]\[{\text{ = }}{\left( {{\text{ - 4}}\sqrt {\text{6}} } \right)^2}\]
$\Rightarrow$\[{\left( {{\text{x - }}\dfrac{1}{{\text{x}}}} \right)^2}\]\[{\text{ = }}{\left( {{\text{ - 4}}} \right)^2}\]\[ \times {\text{ }}{\left( {\sqrt {\text{6}} } \right)^2}\]
By squaring the above values, the square root and square will get cancelled and we have
$\Rightarrow$\[{\left( {{\text{x - }}\dfrac{1}{{\text{x}}}} \right)^2}\]\[{\text{ = }}\] \[{{16 \times 6}}\]
By multiplying we will get
$\Rightarrow$\[{\left( {{\text{x - }}\dfrac{1}{{\text{x}}}} \right)^2}\]\[{\text{ = }}\] \[96\]