Question

Solve the following rational equation and check for extraneous solutions: x/5 = x²/(x+2).

A) x = 0; x equals -1/2 is an extraneous solution.
B) x equals one-half; x = 0 is an extraneous solution.
C) x = 0 and x equals 1/2.
D) x = 0 and x equals negative one-half.

Answer 1

After eliminating the fractions from the equation by multiplying with the least common denominator, simplifying and solving the resultant quadratic equation, we find that the only solution is x = 0. There are no extraneous solutions, as the initial rational equation does not yield any undefined conditions with this solution.

Step-by-step explanation:

To solve the rational equation x/5 = x²/(x+2), we first multiply both sides of the equation by the least common denominator, which is 5(x+2). This will eliminate the fractions and give us a quadratic equation:

5(x²) = x(5)(x+2)

x² = x(x+2)

x² - x(x+2) = 0

x(x - (x+2)) = 0

x(x - x - 2) = 0

x(-2) = 0

After simplifying, we have:

x(0 - 2) = 0

-2x = 0

x = 0

However, when we check to see if this solution makes any denominator zero, we catch that substituting x = 0 into the original equation is not an issue because the denominators will not be zero. Therefore, the only solution is x = 0 and there are no extraneous solutions. The answer is option C: x = 0 and x equals 1/2 is incorrect since 1/2 was not a derived solution from the steps we took.