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Simplifying a product of algebraic fractions and using it to solve a logarithmic equation

Simplifying a product of algebraic fractions and using it to solve a logarithmic equation

6 marks
  1. It is given that \(x>5\). Show that the product \(\dfrac{x^2-9}{x^2-3x-10}\times\dfrac{x+2}{x-3}\) simplifies to \(\dfrac{x+3}{x-5}\).[2]
  2. Using your answer to part (a), or otherwise, solve, for \(x>5\), the equation \(\log_3\left[(x^2-9)(x+2)\right]=2+\log_3\left[(x^2-3x-10)(x-3)\right]\).[4]

Solution:

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Answer:(a) \(\dfrac{(x-3)(x+3)}{(x-5)(x+2)}\times\dfrac{x+2}{x-3}=\dfrac{x+3}{x-5}\) (b) \(x=6\)

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