Apr 13, 2013 Thread starter Admin #1 anemone MHB POTW Director Staff member Feb 14, 2012 3,967 Simplify \(\displaystyle \frac{x^2-4x+3+(x+1)\sqrt{x^2-9}}{x^2+4x+3+(x-1)\sqrt{x^2-9}}\) where \(\displaystyle x>3\).
Simplify \(\displaystyle \frac{x^2-4x+3+(x+1)\sqrt{x^2-9}}{x^2+4x+3+(x-1)\sqrt{x^2-9}}\) where \(\displaystyle x>3\).
Apr 13, 2013 Admin #2 M 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: \(\displaystyle \sqrt{\frac{x-3}{x+3}}\)
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: \(\displaystyle \sqrt{\frac{x-3}{x+3}}\)
Apr 13, 2013 Thread starter Admin #3 anemone MHB POTW Director Staff member Feb 14, 2012 3,967 Bravo, Mark!