Can someone help me with complex fractions and finding the LCD
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Can someone help me with complex fractions and finding the LCD

[From: ] [author: ] [Date: 12-04-27] [Hit: ]
.. but how do I do that? And then do I multiply by that number?Please help!To divide that into the top fraction,......
The problem I'm trying to simplify is:

x^2-2x-8/x^2-9x+20
________________

x^2-4/x^2 -6x+5

I know it looks a little messy but it's the best I can do for writing it on the computer. So I think I'm supposed to find the lowest common denominator of the x^2-9x+20 and the x^2 -6x+5... but how do I do that? And then do I multiply by that number?

Please help!

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Factor everything first to see what you're dealing with:

Top fraction:

x^2 - 2x - 8 = (x + 2)(x - 4)

x^2 - 9x + 20 ==> (x - 4)(x - 5)

This simplifies the top fraction to (x + 2) / (x - 5)

Bottom fraction:

x^2 - 4 ==> (x + 2)(x - 2)

x^2 - 6x + 5 ==> (x - 1)(x - 5)

The bottom fraction is:

[(x + 2)(x - 2)] / [(x - 1)(x - 5)]

To divide that into the top fraction, invert it and multiply:

[(x + 2) / (x - 5)] * [(x - 1)(x - 5) / (x + 2)(x - 2)]

Multiply the numerators and the denominators:

[(x + 2)(x - 1)(x - 5)] / [(x - 5)(x + 2)(x - 2)]

Factoring out (canceling) the (x + 2) and (x - 5) terms leaves you with:

(x - 1) / (x - 2)

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Factor all the polynomial expressions:

( (x - 4)(x + 2)/(x - 5)(x - 4)) / ((x - 2)(x + 2) / (x - 5)(x - 1))

The numerator can be simplified: ((x + 2)/(x - 5)) / (x + 2)(x - 2)/(x - 5)(x - 1)

The LCD is (x - 5)(x - 1)

Multiply both numerator and denominator by LCD and simplify

((x + 2)(x - 5)(x - 1)/(x - 5)) / ((x + 2)(x - 2)(x - 5)(x - 1)/((x - 5)(x - 1))

(x + 2)(x - 1) / (x + 2)(x - 2) --> (x - 1)/(x - 2)

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Probably better to factor and multiply by the reciprocal

x^2-2x-8 = (x - 4)(x + 2)
x^2-9x+20 = (x - 5)(x - 4)
x^2-4 = (x - 2)(x +- 2)
x^2 -6x+5 = (x - 5)(x - 1)

so (x - 4)(x + 2)/ (x - 5)(x - 4) divided by (x - 2)(x + 2)/(x - 5)(x - 1) becomes
(x - 4)(x + 2)/ (x - 5)(x - 4) times (x - 5)(x - 1)/(x - 2)(x + 2)
cancel like factor (x -4)(x -5)(x + 2)
You are left with (x -1)/(x -2)

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x^2-2x-8 = (x - 4)(x + 2)
x^2-9x+20 = (x - 4)(x - 5)
x^2-4 = (x + 2)(x - 2)
x^2 -6x+5 = (x - 5)(x - 1)

Invert and multiply using the factored form of the quadratics
(x - 4)(x + 2)(x - 5)(x - 1) / (x - 4)(x - 5)(x + 2)(x - 2)

Now cancel like factors
(x + 1) / (x - 2)
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