If x + 1/x = 7, then what is the value of x^2 + 1/x^2? | Mathematics MCQ | Examoogle

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If x + 1/x = 7, then what is the value of x² + 1/x²?

Correct Answer

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Option A

Quick Summary:

For $x + \frac{1}{x} = k$ , the value of $x^2 + \frac{1}{x^2}$ is always $k^2 - 2$ . Just compute $7^2 - 2 = 47$ .

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📐MAMath SolutionDirect Formula

📋 Given

It is given that x + 1/x = 7.

🔢 Formula Used

(a+b)2=a2+b2+2ab(a + b)^2 = a^2 + b^2 + 2ab(a+b)2=a2+b2+2ab

⚡ Exam Hall Shortcut / Speed Trick

For $x + \frac{1}{x} = k$ , the value of $x^2 + \frac{1}{x^2}$ is always $k^2 - 2$ . Just compute $7^2 - 2 = 47$ .

⚠️ Common Student Trap / Pitfall

Many students mistakenly square the value 7 to get 49 and forget to subtract 2, resulting in the incorrect answer 49.

📊 Diagram / Illustration

🔢 Step-by-Step Solution

Square both sides

Squaring the equation $x + \frac{1}{x} = 7$ on both sides to involve the squared terms.

(x+1x)2=72(x + \frac{1}{x})^2 = 7^2(x+x1​)2=72

Expand using identity

Apply the identity $(a+b)^2 = a^2 + b^2 + 2ab$ where $a=x$ and $b=1/x$ .

x2+1x2+2(x)(1x)=49x^2 + \frac{1}{x^2} + 2(x)(\frac{1}{x}) = 49x2+x21​+2(x)(x1​)=49

Simplify and solve

Since $x \times (1/x) = 1$ , the equation simplifies to $x^2 + 1/x^2 + 2 = 49$ . Subtracting 2 from both sides gives the final value.

x2+1x2=49−2=47x^2 + \frac{1}{x^2} = 49 - 2 = 47x2+x21​=49−2=47

✅

A is correct because applying the identity results in $49 - 2 = 47$ .

Core Concepts Used

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Algebraic Identities Binomial Expansion Quadratic Relationships

💡 EXAM TIP

This concept is a subset of the 'Algebraic Identities' topic which frequently appears in competitive exams like SSC, Banking, and CAT. Remembering $k^2 - 2$ saves critical time for higher power variations like $x^4 + 1/x^4$ .

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