The problem I'm trying to simplify is:
x^2-2x-8/x^2-9x+20
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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!
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)
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)
( (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)
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)
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)