If the roots of the quadratic equation `3x^(2) - 5x + p = 0` are real and unequal. Then which one of the following is correct ?

Options

A. p = 25/12
B. `p lt 25//12`
C. `p gt 25 //12`
D. `p le 25//12`

Answer

To determine the conditions under which the roots of the quadratic equation \(3x^2 - 5x + p = 0\) are real and unequal, we need to analyze the discriminant of the equation.

### Step-by-Step Solution:

1. **Identify the coefficients**:
The given quadratic equation is in the standard form \(ax^2 + bx + c = 0\), where:
- \(a = 3\)
- \(b = -5\)
- \(c = p\)

2. **Write the discriminant formula**:
The discriminant \(D\) of a quadratic equation is given by:
\[ D = b^2 - 4ac \]

3. **Substitute the values of \(a\), \(b\), and \(c\)**:
Substitute the values of \(a\), \(b\), and \(c\) into the discriminant formula:
\[ D = (-5)^2 - 4 \cdot 3 \cdot p \]
Simplifying this gives:
\[ D = 25 - 12p \]

4. **Set the condition for real and unequal roots**:
For the roots to be real and unequal, the discriminant must be greater than zero:
\[ D > 0 \]
Therefore, we have:
\[ 25 - 12p > 0 \]

5. **Solve the inequality**:
Rearranging the inequality gives:
\[ 25 > 12p \]
Dividing both sides by 12:
\[ p < \frac{25}{12} \]

6. **Conclusion**:
The condition for the roots of the quadratic equation \(3x^2 - 5x + p = 0\) to be real and unequal is:
\[ p < \frac{25}{12} \]