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Simplify (x-5)/(6) - (6)/(x-5)

Mathematics

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Answer

x2−10x−116(x−5)\frac{x^2 - 10x - 11}{6(x-5)}

Step-by-step solution

4 steps

  1. Find a common denominator for the two fractions.

    The common denominator is 6(x−5)6(x-5).

  2. Rewrite each fraction with the common denominator.

    x−56−6x−5=(x−5)(x−5)6(x−5)−6×66(x−5)\frac{x-5}{6} - \frac{6}{x-5} = \frac{(x-5)(x-5)}{6(x-5)} - \frac{6 \times 6}{6(x-5)} =(x−5)26(x−5)−366(x−5)= \frac{(x-5)^2}{6(x-5)} - \frac{36}{6(x-5)}

  3. Combine the fractions by subtracting the numerators.

    =(x−5)2−366(x−5)= \frac{(x-5)^2 - 36}{6(x-5)}

  4. Expand the numerator and simplify.

    Recall that (a−b)2=a2−2ab+b2(a-b)^2 = a^2 - 2ab + b^2.

    (x−5)2=x2−2(x)(5)+52=x2−10x+25(x-5)^2 = x^2 - 2(x)(5) + 5^2 = x^2 - 10x + 25

    Substitute this back into the numerator:

    x2−10x+25−366(x−5)\frac{x^2 - 10x + 25 - 36}{6(x-5)} =x2−10x−116(x−5)= \frac{x^2 - 10x - 11}{6(x-5)}

    The simplified expression is:

    x2−10x−116(x−5)\frac{x^2 - 10x - 11}{6(x-5)}
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