If $a + \frac{1}{a} = 7$, then $a^2 + \frac{1}{a^2} = ?$

Question

Examoogle · Accepted Answer

The correct answer is Option C: 47. For $x + \frac{1}{x} = n$, the value of $x^2 + \frac{1}{x^2}$ is always $n^2 - 2$. Here, $7^2 - 2 = 49 - 2 = 47$.

Step-by-Step Solution & Concepts:
Shortcut Trick: For $x + \frac{1}{x} = n$, the value of $x^2 + \frac{1}{x^2}$ is always $n^2 - 2$. Here, $7^2 - 2 = 49 - 2 = 47$.

Formula Used: $$(a + \frac{1}{a})^2 = a^2 + \frac{1}{a^2} + 2$$

Given: If $a + \frac{1}{a} = 7$, then find the value of $a^2 + \frac{1}{a^2}$.

Steps: Start with the given equation: We are given the expression $a + \frac{1}{a} = 7$.. Square both sides: Square both sides of the equation to introduce the squared terms using the identity $(x+y)^2 = x^2 + y^2 + 2xy$.. Expand the expression: Expanding the left side, we get $a^2 + (\frac{1}{a})^2 + 2 \cdot a \cdot \frac{1}{a} = 49$. Since $a \cdot \frac{1}{a} = 1$, the expression simplifies to $a^2 + \frac{1}{a^2} + 2 = 49$.. Final Calculation: Subtract 2 from both sides to find the target value.

Answer: C is correct because the identity $(a + \frac{1}{a})^2 = a^2 + \frac{1}{a^2} + 2$ results in $49 - 2 = 47$.

Exam Tip: This identity technique is frequently used in competitive exams to quickly solve problems involving reciprocal variables in quadratic and cubic forms.

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