Question

Which of the following statements are true about the given rational equation? Check all of the boxes that apply.

4
x + 6
+
1
x2
=
x + 10
x3 + 6x2

x = 1 is a solution.

x = 0 is a solution.

x = –1 is a solution.

x = –6 is a solution.

Answer 1

Equation: 4/(x+6) + 1/x^2 = (x+10)/(x^3+6x^2) with x^3+6x^2 = x^2(x+6).

Multiply both sides by x^2(x+6) (valid only if x ≠ 0 and x ≠ −6):
4x^2 + (x+6) = x+10 → 4x^2 + 6 = 10 → 4x^2 = 4 → x^2 = 1 → x = ±1.

Both x = 1 and x = −1 are in the domain, and both satisfy the original equation (quick check: x=1 gives 11/7; x=−1 gives 9/5). x = 0 and x = −6 are not allowed (denominators zero), so they are not solutions.

True: x = 1, x = −1.
False: x = 0, x = −6.