Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 3: Problem 218

If \(\alpha, \beta\) are roots of \(8 \mathrm{x}^{2}-3 \mathrm{x}+27=0\) then \(\left(\alpha^{2} / \beta\right)^{(1 / 3)}+\left(\beta^{2} / \alpha\right)^{(1 / 3)}=\) (a) \((1 / 3)\) (b) \((7 / 2)\) (c) 4 (d) \((1 / 4)\)

Short Answer

Expert verified
The sum \(\left(\alpha^{2} / \beta\right)^{1/3} + \left(\beta^{2} / \alpha\right)^{1/3}\) is an imaginary number, so none of the given options (a), (b), (c) or (d) is the correct answer for this exercise.

Step by step solution

01

Identify the Quadratic Equation and Roots

Firstly, we are given the quadratic equation \(8x^{2}-3x+27=0\) with roots α and β.
02

Use Vieta's Formulas for the Sum and Product of Roots

We know that for a quadratic equation \(ax^{2}+bx+c=0\), the sum of the roots is given by \(-b/a\) and the product of the roots by \(c/a\). So for our equation, the sum and product of the roots are: $$\alpha + \beta = \frac{-(-3)}{8} = \frac{3}{8}$$ $$\alpha \beta = \frac{27}{8}$$
03

Multiply the given expression by its conjugate

We want to find the value of \(\left(\alpha^{2} / \beta\right)^{1/3} + \left(\beta^{2} / \alpha\right)^{1/3}\). To make the expression easier to work with, let's multiply the expression by the conjugate of the given expression: $$\left(\left(\alpha^{2} / \beta\right)^{1/3} + \left(\beta^{2} / \alpha\right)^{1/3}\right)\left(\left(\alpha^{2} / \beta\right)^{1/3} - \left(\beta^{2} / \alpha\right)^{1/3}\right)$$
04

Simplify the expression

Now let's simplify the expression we got in step 3: $$\left(\frac{\alpha^{2}}{\beta}\right) - \left(\frac{\beta^{2}}{\alpha}\right) = \frac{\alpha^{3}\beta - \alpha\beta^{3}}{\alpha\beta} = \frac{(\alpha^{3} - \beta^{3})}{\alpha\beta}$$
05

Use the sum and product of roots

Now, let's substitute the relationship we found in step 2 into the expression we got in step 4: \[\frac{(\alpha+\beta)(\alpha^2-\alpha\beta+\beta^2)}{\alpha\beta} =\frac{\frac{3}{8}\left[(\alpha+\beta)^2-3\alpha\beta\right]}{\frac{27}{8}} = \frac{3[(\frac{3}{8})^2 - \frac{9}{8}\frac{27}{8}]}{27}\]
06

Calculate the desired value

Now let's calculate the value of the expression above: \[\frac{3\left[(\frac{9}{64}) - \frac{243}{64}\right]}{27} = \frac{3(\frac{-234}{64})}{27} = \frac{-234}{64} \cdot \frac{1}{9} = -\frac{26}{8} = \frac{-13}{4}\]
07

Find the value of the initial expression

Since we multiplied our initial expression by its conjugate, we need to find the square root of the result we got in step 6 to get the answer for the initial expression. As the result is negative, the sum \(\left(\alpha^{2} / \beta\right)^{1/3} + \left(\beta^{2} / \alpha\right)^{1/3}\) is an imaginary number, which means none of the given options (a), (b), (c) or (d) is the correct answer for this exercise.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

The solution set of equation \(3\left[x^{2}+\left(1 / x^{2}\right)\right]+16[x+(1 / x)]+26=0\) is (a) \([-1,\\{(-1) / 3\\},-3]\) (b) \([\overline{1,(1 / 3), 3}]\) (c) \([-1,(1 / 3), 3]\) (d) \([1,\\{(-1) / 3\\}, 3]\)

The number of real solution of the equation \(\left(1 / 27^{\mathrm{x}}\right)+\left(1 / 12^{\mathrm{x}}\right)=\left(1 / 2.8^{\mathrm{x}}\right)\) (a) 1 (b) 2 (b) 3 (d) 0

If \(\mathrm{b}^{2}=\) ac, equation \(\mathrm{ax}^{2}+2 \mathrm{bx}+\mathrm{c}=0\) and \(\mathrm{d} \mathrm{x}^{2}+2 \mathrm{ex}+\mathrm{f}=0\) have common roots then \((\mathrm{d} / \mathrm{a}),(\mathrm{e} / \mathrm{b}),(\mathrm{f} / \mathrm{c})\) are in (a) A.P. (b) G.P. (c) H.P. (d) None of these

If the difference of the roots of the equation \(x^{2}-p x+q=0\) is 1 then (a) \(p^{2}+4 q^{2}=(1+2 q)^{2}\) (b) \(\mathrm{q}^{2}+4 \mathrm{p}^{2}=(1+2 \mathrm{q})^{2}\) (c) \(p^{2}-4 q^{2}=(1+2 q)^{2}\) (d) \(q^{2}+4 p^{2}=(1-2 q)^{2}\)

The sum of the roots of the equation \(\mathrm{x}^{2}-3|\mathrm{x}|-10=0\) is (a) 3 (b) \(-3\) (c) \(-10\) (d) 0
See all solutions

Recommended explanations on English Textbooks

Cues and Conventions

Read Explanation

Essay Prompts

Read Explanation

Graphology

Read Explanation

Creative Story

Read Explanation

Essay Writing Skills

Read Explanation

Listening and Speaking

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free