If $ a + \frac{1}{a} = 4 $, then $ a^2 + \frac{1}{a^2} = ? $ | Mathematics MCQ

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If $a + \frac{1}{a} = 4$ , then $a^2 + \frac{1}{a^2} =?$

Correct Answer

📐 MA • Math Direct FormulaMathematicsPrevious Year Paper 2024

Option D

Quick Summary:

For the form $x + \frac{1}{x} = n$ , the value of $x^2 + \frac{1}{x^2}$ is $n^2 - 2$ . Here, $4^2 - 2 = 16 - 2 = 14$ .

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📐MAMath SolutionDirect Formula

📋 Given

Given $a + \frac{1}{a} = 4$ .

🔢 Formula Used

(a+1a)2=a2+2(a)(1a)+1a2(a + \frac{1}{a})^2 = a^2 + 2(a)(\frac{1}{a}) + \frac{1}{a^2}(a+a1​)2=a2+2(a)(a1​)+a21​

⚡ Exam Hall Shortcut / Speed Trick

For the form $x + \frac{1}{x} = n$ , the value of $x^2 + \frac{1}{x^2}$ is $n^2 - 2$ . Here, $4^2 - 2 = 16 - 2 = 14$ .

⚠️ Common Student Trap / Pitfall

Many students forget to subtract 2 from the square of the given number and incorrectly choose 16 ( $4^2$ ).

📊 Diagram / Illustration

🔢 Step-by-Step Solution

Apply algebraic identity

Squaring the expression $a + \frac{1}{a} = 4$ on both sides using the identity $(x+y)^2 = x^2 + 2xy + y^2$ .

(a+1a)2=42(a + \frac{1}{a})^2 = 4^2(a+a1​)2=42

Expand the equation

Expand the left side: $a^2 + 2(a)(\frac{1}{a}) + \frac{1}{a^2} = 16$ . Note that $a \times \frac{1}{a} = 1$ .

a2+2+1a2=16a^2 + 2 + \frac{1}{a^2} = 16a2+2+a21​=16

Solve for the final expression

Subtract 2 from both sides to isolate $a^2 + \frac{1}{a^2}$ .

a2+1a2=16−2=14a^2 + \frac{1}{a^2} = 16 - 2 = 14a2+a21​=16−2=14

✅

D is correct because $a^2 + \frac{1}{a^2} = 4^2 - 2 = 14$ .

Core Concepts Used

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Algebraic Identities Square of a Binomial Reciprocal Relationships

💡 EXAM TIP

This concept is a foundational part of the 'Algebra' module and frequently appears in SSC/RRB exams to test quick mental arithmetic using the shortcut $n^2 \pm 2$ .

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