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GB-40-02-001 · Quadratic algebraic fractions

Simplify x2+11x+18x2−4\frac{x^{2} + 11x + 18}{x^2-4}, where x ≠ ±2.

Algebraic Fractions: Simplifying Quadratics Booklet p. 5, p. 6 & p. 7
Method
1) Factorise quadratic numerator using ac/grid and denominator using DOTS
2) Cancel the common bracket factor
Worked Example (GB-40-02-001)
1) Factorise numerator using 2×2 Grid (Booklet p. 5 & p. 7):
×xx+2+2
xxx2x^2+2x+2x
+9+9+9x+9x+18+18
a×c=1×(18)=18a \times c = 1 \times (18) = 18
+1+1+18+18
+2+2+9+9
+3+3+6+6
c is positive so both signs are the same (+ and +)...
+2+9=+11+2 + 9 = +11 ✔
Brackets: (x+9)(x+2)\text{Brackets: } \mathbf{(x + 9)(x + 2)}
2) Factorise denominator using DOTS (Booklet p. 6): x2−4=(x−2)(x+2)x^2 - 4 = (x - 2)(x + 2)
3) Cancel common factor (x+2)(x + 2): (x+2)(x+9)(x−2)(x+2)\quad \frac{\cancel{(x + 2)}(x + 9)}{(x - 2)\cancel{(x + 2)}}
x+9x−2\frac{x+9}{x-2}