Simplifying an algebraic fraction

Algebra

(1MA1) · Higher

ORIGINAL

Factorise top and bottom, then cancel — the standard route through an algebraic fraction.

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Question

Simplify fully

\dfrac{x^2 - 9}{2x^2 + 5x - 3}

Every step worked, with the reasoning.

1
$\dfrac{(x - 3)(x + 3)}{2x^2 + 5x - 3}$
Factorise the numerator as a difference of two squares: $x^2 - 9 = (x-3)(x+3)$ .
2
$\dfrac{(x - 3)(x + 3)}{(2x - 1)(x + 3)}$
Factorise the denominator (product $2 \times -3 = -6$ , sum $+5$ : split as $+6x - x$ ).
3
$\dfrac{x - 3}{2x - 1}$
Cancel the common factor $(x + 3)$ .

Answer: $\dfrac{x - 3}{2x - 1}$

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