$\frac{4 x^{2} - 12 x + 13}{4 x^{2} - 16 x + 15} =$

1 + \frac{1}{2 x - 5} - \frac{2}{2 x - 3}

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1 + \frac{2}{2 x - 5} + \frac{4}{2 x - 3}

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1 + \frac{1}{2 x - 5} + \frac{2}{2 x - 3}

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1 + \frac{4}{2 x - 5} - \frac{2}{2 x - 3}

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Solution

The correct option is C $1 + \frac{4}{2 x - 5} - \frac{2}{2 x - 3}$
$\frac{4 x^{2} - 12 x + 13}{4 x^{2} - 16 x + 15} = \frac{(4 x^{2} - 16 x + 15) + (4 x - 2)}{(4 x^{2} - 16 x + 15)}$
$= 1 + \frac{4 x - 2}{(4 x^{2} - 16 x + 15)}$
$\frac{4 x - 2}{4 x^{2} - 16 x + 15} = \frac{4 x - 2}{(2 x - 5) (2 x - 3)} = \frac{A}{2 x - 5} + \frac{B}{2 x - 3} - - (1)$
The values of A and B will be obtained from (1).
$4 x - 2 = A (2 x - 3) + B (2 x - 5)$
$\Rightarrow 2 A + 2 B = 4$ and $3 A + 5 B = 2$
$\Rightarrow B = - 2$ and $A = 4$
So, $\frac{4 x - 2}{(2 x - 5) (2 x - 3)} = \frac{4}{2 x - 5} - \frac{2}{2 x - 3}$
$\Rightarrow \frac{4 x^{2} - 12 x + 13}{4 x^{2} - 16 x + 15} = 1 + \frac{4 x - 2}{(2 x - 5) (2 x - 3)} = 1 + \frac{4}{(2 x - 5)} - \frac{2}{(2 x - 3)}$

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