Question
upstudy study bank question image url

If the roots of the equation \( 3 x^{2}-5 x+1=0 \) are \( \alpha, \beta \). Find the values of; i. \( \alpha \beta^{2}+\beta \alpha^{2} \) ii. \( \alpha^{2}-\alpha \beta+\beta^{2} \) (Leaving your answer correct to 4 decimal places where appropriate).

Ask by Hanson Bowman. in Ghana
Mar 15,2025

Upstudy AI Solution

Tutor-Verified Answer

Answer

i. \( \alpha \beta^2 + \beta \alpha^2 = 0.5556 \) ii. \( \alpha^2 - \alpha \beta + \beta^2 = 1.7778 \)

Solution

Sign in to Unlock Answers for Free!

A Learning Platform Trusted by Millions of Real Students and Teachers.

star-icon Unlock

Answered by UpStudy AI and reviewed by a Professional Tutor

Extra Insights

To find the values of \( \alpha \beta^{2}+\beta \alpha^{2} \) and \( \alpha^{2}-\alpha \beta+\beta^{2} \), let’s first determine the roots \( \alpha \) and \( \beta \) of the equation \( 3x^{2}-5x+1=0 \). Using the quadratic formula, \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where \( a = 3, b = -5, c = 1 \): 1. Calculate the discriminant: \[ D = (-5)^2 - 4 \cdot 3 \cdot 1 = 25 - 12 = 13 \] 2. Now, applying the quadratic formula: \[ x = \frac{5 \pm \sqrt{13}}{6} \] Thus, the roots are: \[ \alpha = \frac{5 + \sqrt{13}}{6}, \quad \beta = \frac{5 - \sqrt{13}}{6} \] Now we can compute \( \alpha \beta^{2} + \beta \alpha^{2} \): \[ \alpha \beta^{2} + \beta \alpha^{2} = \alpha \beta (\alpha + \beta) \] From Vieta's formulas: - \( \alpha + \beta = \frac{5}{3} \) - \( \alpha \beta = \frac{1}{3} \) Thus, \[ \alpha \beta^{2} + \beta \alpha^{2} = \alpha \beta \cdot (\alpha + \beta) = \frac{1}{3} \cdot \frac{5}{3} = \frac{5}{9} \approx 0.5556 \] Now, for \( \alpha^{2} - \alpha \beta + \beta^{2} \): This can be rewritten using the identity: \[ \alpha^{2} + \beta^{2} = (\alpha + \beta)^2 - 2\alpha\beta \] Using the values from Vieta's formulas: \[ \alpha^{2} + \beta^{2} = \left(\frac{5}{3}\right)^2 - 2 \cdot \frac{1}{3} = \frac{25}{9} - \frac{2}{3} = \frac{25}{9} - \frac{6}{9} = \frac{19}{9} \] Now, substituting back: \[ \alpha^{2} - \alpha \beta + \beta^{2} = \alpha^{2} + \beta^{2} - \alpha \beta = \frac{19}{9} - \frac{1}{3} = \frac{19}{9} - \frac{3}{9} = \frac{16}{9} \approx 1.7778 \] Thus, the final answers are: i. \( \alpha \beta^{2} + \beta \alpha^{2} \approx 0.5556 \) ii. \( \alpha^{2} - \alpha \beta + \beta^{2} \approx 1.7778 \)

Try Premium now!
Try Premium and ask Thoth AI unlimited math questions now!
Maybe later Go Premium
Study can be a real struggle
Why not UpStudy it?
Select your plan below
Premium

You can enjoy

Start now
  • Step-by-step explanations
  • 24/7 expert live tutors
  • Unlimited number of questions
  • No interruptions
  • Full access to Answer and Solution
  • Full Access to PDF Chat, UpStudy Chat, Browsing Chat
Basic

Totally free but limited

  • Limited Solution
Welcome to UpStudy!
Please sign in to continue the Thoth AI Chat journey
Continue with Email
Or continue with
By clicking “Sign in”, you agree to our Terms of Use & Privacy Policy