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Simplify the following:

Simplify the following:-example-1
User Daemone
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5 votes

Answer:


(\left(12x-56\right)x)/(\left(x-4\right)\left(-x^2+20x-32\right))

Explanation:

Simplify


((16)/(x-2)-(4)/(x-4))/((16)/(x)-(x-4)/(x-2))

The first-level numerator is

(16)/(x-4)-(4)/(x-2)\\\\\\= (16(x-4) - 4(x - 2))/((x-2)(x-4))\\\\\\= (16x -64 - 4x + 8)/((x-2)(x-4))\\\\\\= (12x -56)/((x-2)(x-4))

The first-level denominator is

(16)/(x)-(x-4)/(x-2)\\\\\\= (16(x-2) - x(x-4))/(x(x-2))\\\\\\\\= (16x - 32 -x^2 +4x)/(x(x-2))


= (-x^2+20x-32)/(x\left(x-2\right))

Therefore

((16)/(x-2)-(4)/(x-4))/((16)/(x)-(x-4)/(x-2))\\\\\\= (12x -56)/((x-2)(x-4)) /(-x^2+20x-32)/(x\left(x-2\right)) \\\\\\

Use the fraction rule:
(a)/(b) / (d)/(c) = (a)/(b) \cdot (d)/(c)


= (12x -56)/((x-2)(x-4)) \cdot (x(x-2))/(-x^2 +20x -32)}

The (x-2) term cancels out resulting in:

(\left(12x-56\right)x)/(\left(x-4\right)\left(-x^2+20x-32\right))

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