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

1 Answer
Apr 16, 2017

=(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)