因式分解 $27x^2 - 12$ 。由此，化简 $\dfrac{5}{27x^2 - 12} + \dfrac{2}{2 - 3x}$ 。

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完整解法

1. Extract the common factor from both terms.

$27x^2-12=3(9x^2-4)$

2. Factorise the remaining expression using difference of two squares.

$3(9x^2-4)=3\left((3x)^2-2^2\right)=3(3x+2)(3x-2)$

答案

$3(3x+2)(3x-2)$

1. Substitute the factorised form from part (a) and make the second denominator compatible.

From part (a),
$27x^2-12=3(3x+2)(3x-2).$
Also,
$2-3x=-(3x-2).$
So
$\frac{5}{27x^2-12}+\frac{2}{2-3x}=\frac{5}{3(3x+2)(3x-2)}-\frac{2}{3x-2}.$

2. Express both fractions over a common denominator.

The common denominator is
$3(3x+2)(3x-2).$
Hence,

\frac{5}{3(3x+2)(3x-2)}-\frac{2}{3x-2} =\frac{5}{3(3x+2)(3x-2)}-\frac{6(3x+2)}{3(3x+2)(3x-2)}.

3. Combine and simplify the numerator.

\frac{5}{3(3x+2)(3x-2)}-\frac{6(3x+2)}{3(3x+2)(3x-2)} =\frac{5-6(3x+2)}{3(3x+2)(3x-2)}

$=\frac{5-18x-12}{3(3x+2)(3x-2)}$
$=\frac{-18x-7}{3(3x+2)(3x-2)}$
$=-\frac{18x+7}{3(3x+2)(3x-2)}.$

答案

$\displaystyle -\frac{18x+7}{3(3x+2)(3x-2)}$

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