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(x^(2))/(x-2)+(4)/(2-x) Which of the following is equivalent to the given expression shown for x!=2 ? Choose 1 answer: (A) x+2 (B) x-2 (C) (x^(2)-4)/(2-x) (D) (x^(2)+4)/(x-2)

x2x−2+42−x \frac{x^{2}}{x-2}+\frac{4}{2-x} \newlineWhich of the following is equivalent to the given expression shown for x≠2 x \neq 2 ?\newlineChoose 11 answer:\newline(A) x+2 x+2 \newline(B) x−2 x-2 \newline(C) x2−42−x \frac{x^{2}-4}{2-x} \newline(D) x2+4x−2 \frac{x^{2}+4}{x-2}

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Full solution

Q. x2x−2+42−x \frac{x^{2}}{x-2}+\frac{4}{2-x} \newlineWhich of the following is equivalent to the given expression shown for x≠2 x \neq 2 ?\newlineChoose 11 answer:\newline(A) x+2 x+2 \newline(B) x−2 x-2 \newline(C) x2−42−x \frac{x^{2}-4}{2-x} \newline(D) x2+4x−2 \frac{x^{2}+4}{x-2}
  1. Rewrite common denominator: We have the expression (x2)/(x−2)+4/(2−x)(x^2)/(x-2) + 4/(2-x). Notice that the denominators (x−2)(x-2) and (2−x)(2-x) are opposites of each other. We can rewrite 2−x2-x as −(x−2)-(x-2) to have a common denominator.
  2. Combine fractions: Rewrite the second term with the common denominator: 42−x\frac{4}{2-x} becomes −4x−2-\frac{4}{x-2}.
  3. Combine numerators: Now, combine the two fractions with the common denominator: (x2)/(x−2)−4/(x−2)(x^2)/(x-2) - 4/(x-2).
  4. Factor numerator: Combine the numerators over the common denominator: (x2−4)/(x−2)(x^2 - 4)/(x-2).
  5. Factor difference of squares: Notice that x2−4x^2 - 4 is a difference of squares, which can be factored into (x+2)(x−2)(x+2)(x-2).
  6. Factor numerator: Factor the numerator: (x2−4)(x^2 - 4) becomes (x+2)(x−2)(x+2)(x-2).
  7. Cancel common factor: Now the expression is (x+2)(x−2)x−2\frac{(x+2)(x-2)}{x-2}.
  8. Simplified expression: Cancel out the common factor (x−2)(x-2) from the numerator and the denominator.
  9. Simplified expression: Cancel out the common factor (x−2)(x-2) from the numerator and the denominator.The simplified expression is x+2x+2.

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