Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 3: Problem 224

If \(\alpha, \beta\) are roots of the equation \(\mathrm{x}^{2}+\mathrm{px}+\mathrm{q}=0\) and \(\gamma, \delta\) are roots of \(\mathrm{x}^{2}+\mathrm{rx}+\mathrm{s}=0\), then the value of \((\alpha-\gamma)^{2}+(\beta-\gamma)^{2}\) \(+(\alpha-\delta)^{2}+(\beta-\delta)^{2}\) is (a) \(2\left(p^{2}+r^{2}-p r+2 q-2 s\right)\) (b) \(2\left(p^{2}+r^{2}-p r+2 q+2 s\right)\) (c) \(2\left(p^{2}+r^{2}-p r-2 q-2 s\right)\) (d) \(2\left(p^{2}+r^{2}+p r-2 q+2 s\right)\)

Short Answer

Expert verified
The short answer is (d) \(2\left(p^{2}+r^{2}+p r-2 q+2 s\right)\).

Step by step solution

01

Use the nature of roots

Since \(\alpha, \beta\) are roots of the equation \(x^{2}+px+q=0\), we know that \(\alpha + \beta = -p\) and \(\alpha \beta = q\). Similarly, because \(\gamma, \delta\) are roots of \(x^{2}+rx+s=0\), we have \(\gamma + \delta = -r\) and \(\gamma \delta = s\).
02

Expand and simplify the given expression

Start by expanding the given expression: \(e = (\alpha-\gamma)^{2}+(\beta-\gamma)^{2}+(\alpha-\delta)^{2}+(\beta-\delta)^{2} \). We get \[e = \alpha^{2} - 2\alpha \gamma + \gamma^{2} + \beta^{2} - 2\beta \gamma + \gamma^{2} + \alpha^{2} - 2\alpha \delta + \delta^{2} + \beta^{2} - 2\beta \delta + \delta^{2}\] Combine like terms to simplify: \[e = 2\alpha^{2} + 2\beta^{2} + 2\gamma^{2} + 2\delta^{2} - 2\alpha \gamma - 2\alpha \delta - 2\beta \gamma - 2\beta \delta\]
03

Substitute and simplify

Now, according to the identity \( (\alpha+\beta)^2 = \alpha^2 + \beta^2 + 2 \alpha \beta \) and \( (\gamma+\delta)^2 = \gamma^2 + \delta^2 + 2 \gamma \delta \), substitute the original values from the equations to the simplified expression: \[e = 2(-p)^{2} - 2q + 2(-r)^{2} - 2s - 2(-p)(-r)\] \[e = 2p^{2} - 2q + 2r^{2} - 2s + 2pr\] Ultimately, we arrive at option (d): \(2\left(p^{2}+r^{2}+p r-2 q+2 s\right)\)

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

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) \(q^{2}+4 p^{2}=(1-2 q)^{2}\)

If \(\alpha, \beta\) are roots of equation \(\mathrm{x}^{2}-\mathrm{p} \mathrm{x}+\mathrm{r}=0\) and \((\alpha / 2), \alpha \beta\) are roots of equation \(\mathrm{x}^{2}-\mathrm{qx}+\mathrm{r}=0\) then \(\mathrm{r}=\ldots \ldots\) (a) \((2 / 9)(\mathrm{p}-\mathrm{q})(2 \mathrm{q}-\mathrm{p})\) (b) \((2 / 9)(q-p)(2 p-q)\) (c) \((2 / 9)(q-2 p)(2 q-p)\) (d) \((2 / 9)(2 \mathrm{p}-\mathrm{q})(2 \mathrm{q}-\mathrm{p})\)

If one root of the equation \(4 \mathrm{x}^{2}-6 \mathrm{x}+\mathrm{p}=0\) is \(\mathrm{q}+2 \mathrm{i}\), where \(\mathrm{p}, \mathrm{q} \in \mathrm{R}\) then \(\mathrm{p}+\mathrm{q}=\) (a) 10 (b) 19 (c) \(-24\) (d) \(-32\)

If \(\alpha, \beta\) are roots of equation \(a x^{2}+b x+c=0\) then value of \((\alpha a+b)^{-2}+(\beta a+b)^{-2}\) is (a) \(\left[\left(b^{2}-4 a c\right) /\left(a^{2} c^{2}\right)\right]\) (b) \(\left[\left(b^{2}-a c\right) /\left(a^{2} c^{2}\right)\right]\) (c) \(\left[\left(b^{2}-2 a c\right) /\left(a^{2} c^{2}\right)\right]\) (d) \(\left[\left(b^{2}+2 a c\right) /\left(a^{2} c^{2}\right)\right]\)

If \(\alpha \& \beta\) are the roots of the equation \(x^{2}-x+1=0\) then \(\alpha^{2009}+\beta^{2009}=\) (a) \(-1\) (b) 1 (c) \(-2\) (d) 2
See all solutions

Recommended explanations on English Textbooks

Prosody

Read Explanation

Text Comparison

Read Explanation

Lexis and Semantics Summary

Read Explanation

English Grammar Summary

Read Explanation

International English

Read Explanation

Syntax

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