How to determine all real values of a if the equation 2lg(x-1)=lg(ax-3) has a single real solution?

Expert Answers

An illustration of the letter 'A' in a speech bubbles

The equation 2*lg(x-1) = lg (ax-3) has two equal roots.

2*lg(x-1) = lg (ax-3)

=> lg ( x-1)^2 = lg (ax - 3)

taking the antilog of both the sides

=> (x - 1)^2 = ax - 3

=> x^2 - 2x + 1 = ax - 3

=> x^2 - x(2 + a) + 4 = 0

As this quadratic equation has two equal and real roots

(2 + a)^2 - 4*1*4 = 0

=> 4 + a^2 + 4a - 16 = 0

=> a^2 + 4a - 12 = 0

=> a^2 + 6a - 2a - 12 = 0

=> a(a + 6) - 2( a + 6) = 0

=> (a - 2)(a + 6) = 0

=> a = 2 and a = -6

The required values of a are 2 and -6.

Approved by eNotes Editorial Team

Posted on

Soaring plane image

We’ll help your grades soar

Start your 48-hour free trial and unlock all the summaries, Q&A, and analyses you need to get better grades now.

  • 30,000+ book summaries
  • 20% study tools discount
  • Ad-free content
  • PDF downloads
  • 300,000+ answers
  • 5-star customer support
Start your 48-Hour Free Trial