Solve the rational equation.

x2+6

x−1
+
x−2

x−1
=2x

If you have multiple answers, separate them with a comma. Reduce all fractions. Enter fractions using "/".

Question

Solve the rational equation.

x2+6

x−1
+
x−2

x−1
=2x

If you have multiple answers, separate them with a comma.   Reduce all fractions.  Enter fractions using "/".

BlackTom AI · Accepted Answer

We start by restating the problem to ensure clarity: solve the rational equation (x^2 + 6)/(x − 1) + (x − 2)/(x − 1) = 2x, with the understanding that x cannot be 1 because that would make the denominators zero.

First, combine the fractions on the left since they share the same denominator. The sum becomes [ (x^2 + 6) + (x − 2) ] / (x − 1) = (x^2 + x + 4) / (x − 1).

Thus the equation is (x^2 + x + 4) / (x − 1) = 2x. To remove the fraction, multiply both sides by (x − 1), while remembering the restriction x ≠ 1. This yields x^2 + x + 4 = 2x(x − 1) = 2x^2 − 2x.

Bring all terms to one side to obtain a standard quadratic: 0 = 2x^2 − 2x − x^2 − x − 4, which simplifies to x^2 − 3x − 4 = 0.

Now factor the quadratic: x^2 − 3x − 4 = (x − 4)(x + 1) = 0. Therefore the potential solutions are x = 4 or x = −1.

Check domain conditions: x ≠ 1, which is satisfied by both 4 and −1, so neither is excluded by the denominators. No other extraneous solutions arise from the multiplication step since we explicitly noted the restriction.

Next, consider the two given candidate solutions. If the problem expects a comma-separated list of all valid solutions, the set of solutions is {−1, 4}. The two orderings you see—"−1,4" and "4,−1"—represent the same solution set; the order does not change correctness, it merely reflects different ways to present the same pair of roots.

Why is the solution x = −1 valid? Substituting back into the original equation yields a true equality after simplifying, and it does not violate any domain restriction. Why is x = 4 valid? The same check confirms it also satisfies the equation and respects the domain.

Why not any other numbers? The quadratic gives exactly these two roots; no additional roots arise from the rational equation beyond those two, assuming x ≠ 1 is kept in mind. The only potential pitfall would be accidentally discarding a domain restriction or mismanaging the cross-multiplication, but those have been properly accounted for here.

In summary, the equation yields two valid roots, −1 and 4, and the listed answers that present those two numbers in either order both convey the correct solution set.

Solve the rational equation. x2+6 x−1 + x−2 x−1 =2x If you have multiple answers, separate them with a comma. Reduce all fractions. Enter fractions using "/".Short answer

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