Question

If $x = 7 - 4\sqrt 3 $ , the value of ${x^2} + \dfrac{1}{{{x^2}}}$ will be
A. 146
B. 148
C. 194
D. 196

Answer 1

Hint:For solving this particular problem , use the given equation . find the reciprocal of the given statement , simplify the equation , rationalise the equation and then substitutes these value to find the value of ${x^2} + \dfrac{1}{{{x^2}}}$ .

Complete solution step by step:
It is given that ,
$x = 7 - 4\sqrt 3 $ (given)
Therefore , we can say that ,
$\dfrac{1}{x} = \dfrac{1}{{7 - 4\sqrt 3 }} \times \dfrac{{7 + 4\sqrt 3 }}{{7 + 4\sqrt 3 }}$
$
= \dfrac{{7 - 4\sqrt 3 }}{{49 - 48}} \\
= 7 + 4\sqrt 3 \\
$
Now ,
$ \Rightarrow x + \dfrac{1}{x} = (7 - 4\sqrt 3 ) + (7 + 4\sqrt 3 )$
$ = 14$
Now ,
$ \Rightarrow {\left( {x + \dfrac{1}{x}} \right)^2} = {(14)^2}$
$
\Rightarrow {x^2} + \dfrac{1}{{{x^2}}} + 2 = 196 \\
\Rightarrow {x^2} + \dfrac{1}{{{x^2}}} = 196 - 2 \\
$
$ = 194$
Hence we get our desired solution.
And we can say that option C is the correct option.

Additional Information:
The expressions are formed by performing operations like addition,
subtraction, multiplication and division on the variables and constants.
•An equation may be a condition on a variable (or variables) specified two expressions within the variable (variables) have equal value.
•The value of the variable that the equation is satisfied is termed the answer or root of the equation.
•An equation remains the same if the LHS and also the RHS are interchanged.
•Transposing means moving from one side to the opposite. When a term is transposed from one side of the equation to the opposite side, its sign gets changed.
•Transposition of an expression is administered within the same way because the transposition of a term.

Note: For simplifying the equation , we rationalise the equation . “Rationalizing the denominator” is after we move a root (like a root or cube root) from the underside of a fraction to the highest. we will multiply both top and bottom by the conjugate of the denominator , which cannot change the worth of the fraction.