Question

If \[{x^2} - \dfrac{1}{{{x^2}}} = - 1\], then find the value of \[{x^2} + \dfrac{1}{{{x^2}}}\]
A) \[\sqrt 2 \]
B) \[2\]
C) \[\sqrt 5 \]
D) None of these

Answer 1

Hint: The given question is in the form of an algebraic equation. We have to solve the given question by using the factorization method. We shall first square the equation given using the formula \[{(a - b)^2} = {a^2} - 2ab + {b^2}\] and then find out the required value.

Complete step by step solution:
A mathematical statement that expresses the relationship between two values is known as an equation. Generally, an equal sign is used to compare the two values. Meanwhile a mathematical statement in which two expressions are set equal to each other is known as an algebraic equation.
There are different types of algebraic equations such as linear, quadratic, cubic, polynomial and trigonometric equations.
Example of algebraic equation is as follows:
\[5x + 10 = 15\] is an example of a linear equation since \[x\] has a power of \[1\].
\[5{x^2} + 5x + 10 = 0\] is an example of a quadratic equation since \[x\] has a power of \[2\].
Similarly, if \[x\] has a power of \[3\] then it is called a cubic equation and with power more than \[3\] is called a polynomial equation.
Let us solve the sum as follows:
\[{x^2} - \dfrac{1}{{{x^2}}} = - 1\]
Squaring on both the sides of equation, we get,
\[{({x^2} - \dfrac{1}{{{x^2}}})^2} = {( - 1)^2}\]
Using the formula \[{(a - b)^2} = {a^2} - 2ab + {b^2}\], we get,
\[{x^4} - 2({x^2})(\dfrac{1}{{{x^2}}}) + \dfrac{1}{{{x^4}}} = 1\]
Simplifying the equation, we get,
\[{x^4} - 2 + \dfrac{1}{{{x^4}}} = 1\]
Taking \[2\]on the right-hand side of equation, we get,
\[{x^4} + \dfrac{1}{{{x^4}}} = 1 + 2\]
\[{x^4} + \dfrac{1}{{{x^4}}} = 3\]
Now we have to add \[2\] on both the sides of equation to form the equation \[{({x^2} + \dfrac{1}{{{x^2}}})^2}\] as follows:
\[{x^4} + \dfrac{1}{{{x^4}}} + 2 = 3 + 2\]
\[{x^4} + \dfrac{1}{{{x^4}}} + 2 = 5\]
Now using the formula \[{(a + b)^2} = {a^2} + 2ab + {b^2}\], we can write the above equation in compact form as follows:
\[{({x^2} + \dfrac{1}{{{x^2}}})^2} = 5\]
Taking square root on both the sides, we get,
\[{x^2} + \dfrac{1}{{{x^2}}} = \sqrt 5 \]
Hence, Option (C) \[\sqrt 5 \] is the correct answer.

Note:
\[{x^4} + \dfrac{1}{{{x^4}}} + 2 = {({x^2} + \dfrac{1}{{{x^2}}})^2}\] is proved as follows using \[{(a + b)^2} = {a^2} + 2ab + {b^2}\]:
Expanding the RHS side of equation, we get,
\[{x^4} + 2({x^2})(\dfrac{1}{{{x^2}}}) + \dfrac{1}{{{x^4}}}\]
Simplifying the equation, we get,
\[{x^4} + 2 + \dfrac{1}{{{x^4}}}\] or
\[{x^4} + \dfrac{1}{{{x^4}}} + 2\]