If x + 1/x = 5, what is the value of x^2 + 1/x^2? | Mathematics MCQ | Examoogle

Examoogle • User • info@examoogle.com • EE-2024-8821

Chapter 1 of 12 • Page 1 of 248🔒 Protected PDF • Watermarked

Back to Practice Questions

MathematicsAlgebra

Prev Next

If x + 1/x = 5, what is the value of x² + 1/x²?

Correct Answer

📐 MA • Math Direct FormulaMathematicsAlgebra

Option A

Quick Summary:

For $x + \frac{1}{x} = n$ , the value of $x^2 + \frac{1}{x^2}$ is always $n^2 - 2$ . Here, $5^2 - 2 = 25 - 2 = 23$ .

🎯

Crack Algebra with Examoogle SuperTeachers

Get Free Demo Classes & Live Mocks →

📐MAMath SolutionDirect Formula

📋 Given

We are given the algebraic expression $x + \frac{1}{x} = 5$ .

🔢 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 $x + \frac{1}{x} = n$ , the value of $x^2 + \frac{1}{x^2}$ is always $n^2 - 2$ . Here, $5^2 - 2 = 25 - 2 = 23$ .

⚠️ Common Student Trap / Pitfall

Students often make a sign error by calculating $n^2 + 2$ instead of $n^2 - 2$ or forgetting to square the number 5 correctly.

📊 Diagram / Illustration

🔢 Step-by-Step Solution

Square both sides of the given equation

To introduce the square terms, we square the given equation $(x + \frac{1}{x}) = 5$ .

(x+1x)2=52(x + \frac{1}{x})^2 = 5^2(x+x1​)2=52

Expand the LHS using the binomial formula

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

x2+1x2+2(x)(1x)=25x^2 + \frac{1}{x^2} + 2(x)(\frac{1}{x}) = 25x2+x21​+2(x)(x1​)=25

Simplify and solve for the target expression

Since $x \cdot \frac{1}{x} = 1$ , the equation simplifies to $x^2 + \frac{1}{x^2} + 2 = 25$ . Subtract 2 from both sides.

x2+1x2=25−2=23x^2 + \frac{1}{x^2} = 25 - 2 = 23x2+x21​=25−2=23

✅

A is correct because squaring the given expression results in $x^2 + 2 + 1/x^2 = 25$ , which simplifies to $23$ .

Core Concepts Used

Click any tag to open in AI Tutor

Algebraic Identities Squaring Binomials Reciprocal Properties

💡 EXAM TIP

This shortcut $n^2 \pm 2$ is extremely useful for higher-order power questions like $x^4 + 1/x^4$ , where you simply repeat the process on the result.

Was this solution helpful?

⚡ Practice on Android App · 4.8 ★

Solve 10,000+ More Questions with AI Tutor

Get instant step-by-step AI explanations, full shift Previous Year Papers (SSC GD, RRB, SSC JE), and regional rank analysis on the official Examoogle App.

Free Download • 100K+ Aspirants

Discussion (0)

Loading discussion...

Prev Next

Correct Answer

Solve 10,000+ More Questions with AI Tutor

Related Questions

Discussion (0)