Solution:

The given equation is 2x³-x²-4x²=0

Let's find the rational roots of the equation using the Rational Root Theorem.

Rational Root Theorem: If a polynomial equation has integer coefficients, then any rational root of the polynomial must have numerator that divides the constant term and denominator that divides the leading coefficient.

In our case, the constant term is 0 and the leading coefficient is 2. Therefore, any rational root of the equation must be of the form p/q, where p is a factor of 0 and q is a factor of 2.

The factors of 0 are ±1, ±2, ±4, ±8, etc.

The factors of 2 are ±1, ±2.

Therefore, the possible rational roots of the equation are as follows:

±1/1, ±2/1, ±4/1, ±8/1, ±1/2, ±2/2, ±4/2, ±8/2

Simplifying the above expression:

±1, ±2, ±4, ±8, ±1/2, ±1, ±2, ±4

Simplifying the above expression again:

±1, ±2, ±4, ±8

Now, we can check each of these possible rational roots by substituting them into the equation and see if they satisfy the equation.

When we substitute x=1/2 into the equation, we get:

2(1/2)³-(1/2)²-4(1/2)² = 1/4 - 1/4 - 1 = -1

Therefore, x=1/2 is not a solution.

When we substitute x=-1/2 into the equation, we get:

2(-1/2)³-(-1/2)²-4(-1/2)² = -1/4 - 1/4 + 1 = 0

Therefore, x=-1/2 is a solution.

When we substitute x=2 into the equation, we get:

2(2)³-(2)²-4(2)² = 16 - 4 - 16 = -4

Therefore, x=2 is not a solution.

When we substitute x=-2 into the equation, we get:

2(-2)³-(-2)²-4(-2)² = -16 - 4 + 16 = -4

Therefore, x=-2 is not a solution.

Hence, the rational root of the equation is x=-1/2.

Therefore, the correct answer is (a) 1/2.