Question:
If
\[
x+\frac1x=3,
\]
then the value of
\[
x^2+\frac1{x^2}
\]
is:
If
\[
x+\frac1x=3,
\]
then the value of
\[
x^2+\frac1{x^2}
\]
is:
Show Hint
Memorize the identity
\[
\left(x+\frac1x\right)^2
=
x^2+\frac1{x^2}+2.
\]
It is one of the most frequently used identities in algebra.
Updated On: Oct 5, 2026
- \(5\)
- \(7\)
- \(9\)
- \(11\)
Show Solution
Verified By Collegedunia
The Correct Option is B
Solution and Explanation
Concept:
Questions involving expressions such as
\[
x+\frac1x
\]
are usually solved using algebraic identities. Instead of finding the value of \(x\), we manipulate the given expression to obtain the required expression.
A very important identity is
\[
\left(x+\frac1x\right)^2
=
x^2+\frac1{x^2}+2.
\]
This identity directly connects the given information with the required quantity.
Step 1: Write the given condition. \[ x+\frac1x=3. \]
Step 2: Square both sides. \[ \left(x+\frac1x\right)^2 = 3^2. \] \[ = 9. \]
Step 3: Apply the standard identity. \[ x^2+\frac1{x^2}+2 = 9. \]
Step 4: Solve for the required expression. Subtracting \(2\) from both sides, \[ x^2+\frac1{x^2} = 9-2. \] \[ = 7. \]
Step 5: Verification. Substituting the obtained value into the identity confirms that \[ 7+2=9. \] The condition is satisfied exactly.
Step 6: Final Conclusion. \[ \boxed{x^2+\frac1{x^2}=7} \] Hence the correct answer is \[ \boxed{\text{Option (B)}}. \]
Step 1: Write the given condition. \[ x+\frac1x=3. \]
Step 2: Square both sides. \[ \left(x+\frac1x\right)^2 = 3^2. \] \[ = 9. \]
Step 3: Apply the standard identity. \[ x^2+\frac1{x^2}+2 = 9. \]
Step 4: Solve for the required expression. Subtracting \(2\) from both sides, \[ x^2+\frac1{x^2} = 9-2. \] \[ = 7. \]
Step 5: Verification. Substituting the obtained value into the identity confirms that \[ 7+2=9. \] The condition is satisfied exactly.
Step 6: Final Conclusion. \[ \boxed{x^2+\frac1{x^2}=7} \] Hence the correct answer is \[ \boxed{\text{Option (B)}}. \]
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