If the roots of the equation `3x^(2) - 5x + q = 0` are equal, then what is the value of q?

Options

A. 2
B. `5//12`
C. `12//25`
D. `25//12`

Answer

To find the value of \( q \) for which the roots of the equation \( 3x^2 - 5x + q = 0 \) are equal, we can follow these steps:

### Step 1: Understand the condition for equal roots
The roots of a quadratic equation \( ax^2 + bx + c = 0 \) are equal when the discriminant \( D \) is equal to zero. The discriminant is given by the formula:
\[ D = b^2 - 4ac \]

### Step 2: Identify the coefficients
In our equation \( 3x^2 - 5x + q = 0 \):
- \( a = 3 \)
- \( b = -5 \)
- \( c = q \)

### Step 3: Set up the equation for the discriminant
Since the roots are equal, we set the discriminant \( D \) to zero:
\[ D = b^2 - 4ac = 0 \]
Substituting the values of \( a \), \( b \), and \( c \):
\[ (-5)^2 - 4 \cdot 3 \cdot q = 0 \]

### Step 4: Calculate the discriminant
Calculating \( (-5)^2 \):
\[ 25 - 12q = 0 \]

### Step 5: Solve for \( q \)
Now, we can solve for \( q \):
\[ 25 = 12q \]
\[ q = \frac{25}{12} \]

### Conclusion
The value of \( q \) for which the roots of the equation \( 3x^2 - 5x + q = 0 \) are equal is:
\[ \boxed{\frac{25}{12}} \]

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