Question

Solve the rational equation 5x​=x+2x2​ and check for extraneous solutions.

A) x=−5,0,5; Extraneous solutions: None
B) x=−5,0,5; Extraneous solutions: x=−2
C) x=−2,0,5; Extraneous solutions: None
D) x=−2,0,5; Extraneous solutions: x=−5

Answer 1

The rational equation 5x = x + 2x^2 is rewritten as 2x^2 - x = 0, which factors to x(2x - 1) = 0, yielding solutions x = 0 and x = 1/2 with no extraneous solutions.

Step-by-step explanation:

To solve the rational equation 5x = x + 2x2, we need to rewrite it in the standard form of a quadratic equation ax2 + bx + c = 0. After rearranging, we get 2x2 - x = 0. This can be factored to x(2x - 1) = 0, giving us two solutions: x = 0 and x = 1/2.

Next, we check for extraneous solutions by substituting the values back into the original equation. Both solutions are valid and therefore we have no extraneous solutions. In conclusion, the correct choice in this context is x = 0 and x = 1/2 with no extraneous solutions.