MHB Simplifying Rational Expression

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The expression $$\frac{x^2-4x+3+(x+1)\sqrt{x^2-9}}{x^2+4x+3+(x-1)\sqrt{x^2-9}}$$ simplifies to $$\sqrt{\frac{x-3}{x+3}}$$ for $$x>3$$. Initial factoring of the numerator and denominator reveals common terms under the radicals. Further simplification leads to the cancellation of common factors. The final result highlights the relationship between the square roots of the expressions. This process demonstrates effective techniques for simplifying rational expressions involving square roots.
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Simplify $$\frac{x^2-4x+3+(x+1)\sqrt{x^2-9}}{x^2+4x+3+(x-1)\sqrt{x^2-9}}$$ where $$x>3$$.
 
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If we factor the first 3 terms in the numerator and denominator, and factor under the radicals, we obtain:

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

Factoring further, we obtain:

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

Dividing out the common factors, we are left with:

$$\sqrt{\frac{x-3}{x+3}}$$
 
Bravo, Mark!
 
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