Welcome to our community

Be a part of something great, join today!

Simplifying Rational Expression

  • Thread starter
  • Admin
  • #1

anemone

MHB POTW Director
Staff member
Feb 14, 2012
3,682
Simplify \(\displaystyle \frac{x^2-4x+3+(x+1)\sqrt{x^2-9}}{x^2+4x+3+(x-1)\sqrt{x^2-9}}\) where \(\displaystyle x>3\).
 

MarkFL

Administrator
Staff member
Feb 24, 2012
13,775
If we factor the first 3 terms in the numerator and denominator, and factor under the radicals, we obtain:

\(\displaystyle \frac{(x-1)(x-3)+(x+1)\sqrt{(x+3)(x-3)}}{(x+1)(x+3)+(x-1)\sqrt{(x+3)(x-3)}}\)

Factoring further, we obtain:

\(\displaystyle \frac{\sqrt{x-3}((x-1)\sqrt{x-3}+(x+1)\sqrt{x+3})}{\sqrt{x+3}((x+1)\sqrt{x+3}+(x-1)\sqrt{x-3})}\)

Dividing out the common factors, we are left with:

\(\displaystyle \sqrt{\frac{x-3}{x+3}}\)
 
  • Thread starter
  • Admin
  • #3

anemone

MHB POTW Director
Staff member
Feb 14, 2012
3,682
Bravo, Mark!