Question

How do you simplify `(x^2-8x+12)/(x^2+7x-18)` and then find the excluded values?

Answer 1

`=(x-6)/(x+9)` `" "` For all `x in (-oo,-9)uu(-9,2)uu(2,+oo)`

Explanation:

To simplify a fraction we think about factorizing the numerator and denominator .
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Factorizing the numerator:
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We think about the trial and error method that says:
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Given`" "color(red)(x^2+Sx+P`
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If we find two real numbers `color(blue)a` and `color(blue)b`
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Such that `S=a+b" "and " " P=axxb`
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Then
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`color(red)(x^2-Sx+P=(x+color(blue)a)(x+color(blue)b))`
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`x^2-8x+12`
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`=(x-2)(x-6)`
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Factorizing the denominator:
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`x^2+7x-18`
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`=(x-2)(x+9)`
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From the above factorization of numerator and denominator we
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recognize the common factor `(x-2)`
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`(x^2 -8x+12)/(x^2+7x-18)`
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`=((x-2)(x-6))/((x-2)(x+9))`
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`x-2!=0rArrx!=2`
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`x+9!=0rArrx!=-9`
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Therefore,
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`(x^2 -8x+12)/(x^2+7x-18)`
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`=(x-6)/(x+9)` `" "` For all `x in (-oo,-9)uu(-9,2)uu(2,+oo)`