Which of the following are the solutions of `2x^(2)-5x-3=0`?
(i) `x=2`
(ii) `x = 3`
(iii) `x=(-1)/(2)`

Options

A. only (i)
B. (i) and (ii)
C. (ii) and (iii)
D. (i) , (ii) and (iii)

Explanation

To determine which of the given values are solutions of the quadratic equation \(2x^2 - 5x - 3 = 0\), we will substitute each value into the equation and check if it satisfies the equation (i.e., if it makes the left-hand side equal to zero).

### Step 1: Substitute \(x = 2\)

Substituting \(x = 2\) into the equation:

\[ 2(2)^2 - 5(2) - 3 = 0 \]

Calculating each term:

\[ 2(4) - 10 - 3 = 0 \]

\[ 8 - 10 - 3 = 0 \]

\[ 8 - 13 = -5 \quad \text{(not equal to 0)} \]

Thus, \(x = 2\) is **not** a solution.

### Step 2: Substitute \(x = 3\)

Now, substituting \(x = 3\):

\[ 2(3)^2 - 5(3) - 3 = 0 \]

Calculating each term:

\[ 2(9) - 15 - 3 = 0 \]

\[ 18 - 15 - 3 = 0 \]

\[ 18 - 18 = 0 \quad \text{(equal to 0)} \]

Thus, \(x = 3\) **is** a solution.

### Step 3: Substitute \(x = -\frac{1}{2}\)

Finally, substituting \(x = -\frac{1}{2}\):

\[ 2\left(-\frac{1}{2}\right)^2 - 5\left(-\frac{1}{2}\right) - 3 = 0 \]

Calculating each term:

\[ 2\left(\frac{1}{4}\right) + \frac{5}{2} - 3 = 0 \]

\[ \frac{1}{2} + \frac{5}{2} - 3 = 0 \]

\[ \frac{6}{2} - 3 = 0 \]

\[ 3 - 3 = 0 \quad \text{(equal to 0)} \]

Thus, \(x = -\frac{1}{2}\) **is** a solution.

### Summary of Results

The solutions of the equation \(2x^2 - 5x - 3 = 0\) from the given options are:
- \(x = 3\) (is a solution)
- \(x = -\frac{1}{2}\) (is a solution)
- \(x = 2\) (is not a solution)