If 2x^4+x^3-11x^2+x+2=0, then the values of x+1/x are:

A. -3, 5/2. Step 1: x=0 is not a root, so divide by x^2: 2(x^2+1x^2)+(x+1/x)-11=0. Step 2: Put t=x+1/x, so x^2+1x^2=t^2-2: 2t^2+t-15=0. Step 3: (t+3)(2t-5)=0, so t=-3 or t=5/2. Check: t=5/2 gives x=2 or 1/2; at x=2: 32+8-44+2+2=0. Trap: Writing x^2+1x^2=t^2 (forgetting the -2) gives 2t^2+t-11=0, whose roots are irrational and match no option.

Question 18 of 34 · Quadratic Equations

Questionmedium

This question was previously asked in

NIMCET 2009

If $2 x^{4} + x^{3} - 11 x^{2} + x + 2 = 0$ , then the values of $x + \frac{1}{x}$ are:

Answer (Detailed Solution Below)View answer ▾

Answer: Option A — $- 3, \frac{5}{2}$

Detailed Solution

$x = 0$ is not a root, so divide by $x^{2}$ : $2 (x^{2} + \frac{1}{x ^{2}}) + (x + \frac{1}{x}) - 11 = 0$ .

Put $t = x + \frac{1}{x}$ , so $x^{2} + \frac{1}{x ^{2}} = t^{2} - 2$ : $2 t^{2} + t - 15 = 0$ .

$(t + 3) (2 t - 5) = 0$ , so $t = - 3$ or $t = \frac{5}{2}$ .

✓ Check

t = \frac{5}{2}

gives

x = 2

\frac{1}{2}

; at

x = 2

32 + 8 - 44 + 2 + 2 = 0

⚠ Common trap

Writing

x^{2} + \frac{1}{x ^{2}} = t^{2}

(forgetting the

- 2

) gives

2 t^{2} + t - 11 = 0

, whose roots are irrational and match no option.

Quick tip

$Divide by x^{2}, t = x + \frac{1}{x} : 2 t^{2} + t - 15 = 0 \Rightarrow t = - 3, \frac{5}{2}$

💬 Discussions on this question

Ask a doubt ›

Still stuck on this question? Ask our faculty and fellow students — answers appear here.

Ask a doubt

Quadratic Equations — Preparation

This question tests your understanding of Quadratic Equations, a key topic in the Quantitative Aptitude section. The detailed solution above uses a step-by-step approach to show the reasoning, and where available, a shortcut that helps you solve similar questions in under 90 seconds during the actual exam.

A quadratic equation has the form ax^2 + bx + c = 0, where a is not zero. This topic covers solving quadratics by factorisation and by the quadratic formula, the nature of roots using the discriminant b^2 - 4ac, the sum (-b/a) and product (c/a) of roots, and forming an equation from its roots. In aptitude exams, a common question type gives two quadratic equations and asks you to compare their roots (whether x > y, x < y and so on). In engineering and MCA entrance exams, questions also test conditions on roots, common roots, and symmetric expressions such as the sum of squares of roots. Using the sum and product of roots is usually far faster than finding the roots themselves. Practise the quadratic equation questions below, each with a step-by-step solution.

‹ All Quadratic Equations questions Browse all topics ›

If $2 x^{4} + x^{3} - 11 x^{2} + x + 2 = 0$ , then the values of $x + \frac{1}{x}$ are:

💬 Discussions on this question

More Quadratic Equations Questions

More Quantitative Aptitude Questions

Quadratic Equations — Preparation

If 2x4+x3−11x2+x+2=0, then the values of x+x1​ are:

💬 Discussions on this question

More Quadratic Equations Questions

More Quantitative Aptitude Questions

Quadratic Equations — Preparation

If $2 x^{4} + x^{3} - 11 x^{2} + x + 2 = 0$ , then the values of $x + \frac{1}{x}$ are: