If 5/3 and 3 are the two roots of a quadratic equation, then which one of the following is the correct quadratic equation?

Options

A. `3x^2- 15x -14 =0`
B. `3x^2 +14x -15 =0`
C. `3x^2 -14x +15 =0`
D. none of these

Answer

To find the quadratic equation given the roots \( \frac{5}{3} \) and \( 3 \), we can follow these steps:

### Step 1: Identify the roots
The roots of the quadratic equation are given as:
- \( A = \frac{5}{3} \)
- \( B = 3 \)

### Step 2: Calculate the sum of the roots
The sum of the roots \( S \) can be calculated as:
\[ S = A + B = \frac{5}{3} + 3 \]
To add these, we need a common denominator. The number 3 can be expressed as \( \frac{9}{3} \):
\[ S = \frac{5}{3} + \frac{9}{3} = \frac{5 + 9}{3} = \frac{14}{3} \]

### Step 3: Calculate the product of the roots
The product of the roots \( P \) is calculated as:
\[ P = A \times B = \frac{5}{3} \times 3 \]
Simplifying this gives:
\[ P = 5 \]

### Step 4: Write the quadratic equation
Using the general form of a quadratic equation given the roots:
\[ x^2 - (S)x + P = 0 \]
Substituting the values of \( S \) and \( P \):
\[ x^2 - \left(\frac{14}{3}\right)x + 5 = 0 \]

### Step 5: Eliminate the fraction by multiplying through by 3
To eliminate the fraction, multiply the entire equation by 3:
\[ 3x^2 - 14x + 15 = 0 \]

### Conclusion
Thus, the required quadratic equation is:
\[ 3x^2 - 14x + 15 = 0 \]

### Final Answer
The correct quadratic equation is \( 3x^2 - 14x + 15 = 0 \).

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