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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

1

Simplify fully x2−92x2+5x−3\dfrac{x^2 - 9}{2x^2 + 5x - 3}.

Every step worked, with the reasoning.

  1. 1
    (x−3)(x+3)2x2+5x−3\dfrac{(x - 3)(x + 3)}{2x^2 + 5x - 3}

    Factorise the numerator as a difference of two squares: x2−9=(x−3)(x+3)x^2 - 9 = (x-3)(x+3).

  2. 2
    (x−3)(x+3)(2x−1)(x+3)\dfrac{(x - 3)(x + 3)}{(2x - 1)(x + 3)}

    Factorise the denominator (product 2×−3=−62 \times -3 = -6, sum +5+5: split as +6x−x+6x - x).

  3. 3
    x−32x−1\dfrac{x - 3}{2x - 1}

    Cancel the common factor (x+3)(x + 3).

Answer: x−32x−1\dfrac{x - 3}{2x - 1}

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