Question

1. Show that the rods of the equation \( (r+1) x^{2}+4 x+1=r \) are real for all real values of r.

Ask by Rose Rogers. in South Africa
Mar 14,2025

Upstudy AI Solution

Tutor-Verified Answer

Answer

For all real values of \( r \), the equation \( (r+1)x^2 + 4x + 1 = r \) always has real solutions because its discriminant is always positive.

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

The Deep Dive

To demonstrate that the rods of the equation \( (r+1) x^{2}+4 x+1=r \) are real for all real values of \( r \), we can rearrange the equation to the standard quadratic form: \[ (r+1)x^2 + 4x + (1 - r) = 0 \] This is a quadratic equation in \( x \), and for the roots to be real, the discriminant must be non-negative. The discriminant \( D \) for a quadratic equation \( ax^2 + bx + c = 0 \) is given by: \[ D = b^2 - 4ac \] Applying this to our equation, where \( a = r + 1 \), \( b = 4 \), and \( c = 1 - r \): \[ D = 4^2 - 4(r + 1)(1 - r) \] \[ D = 16 - 4(r + 1)(1 - r) \] Expanding \( (r + 1)(1 - r) \): \[ (r + 1)(1 - r) = r - r^2 + 1 - r = 1 - r^2 \] Thus, \[ D = 16 - 4(1 - r^2) = 16 - 4 + 4r^2 = 12 + 4r^2 \] Since \( 4r^2 \) is always non-negative and adding 12 keeps it non-negative, we conclude that \( D \geq 12 \). Therefore, the roots of the quadratic equation are real for all real values of \( r \). Thus, the rods represented by this equation are always real regardless of the value of \( r \).

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