Choose the correct option if it is known that one root of the quadratic equation `ax^(2)+bx+c=0` is three times the other root?

Options

A. `b^(2)=3ac`
B. `3b^(2)=16ac`
C. `a^(2)=9b^(2)c`
D. `4c^(2)=9ab`

Answer

To solve the problem, we need to analyze the given quadratic equation \( ax^2 + bx + c = 0 \) with the condition that one root is three times the other root. Let's denote the roots as \( \alpha \) and \( \beta \).

### Step-by-Step Solution:

1. **Define the Roots**:
Let \( \alpha \) be the larger root and \( \beta \) be the smaller root. According to the problem, we have:
\[ \alpha = 3\beta \]

2. **Use the Sum and Product of Roots**:
For a quadratic equation \( ax^2 + bx + c = 0 \):
- The sum of the roots \( \alpha + \beta = -\frac{b}{a} \)
- The product of the roots \( \alpha \beta = \frac{c}{a} \)

3. **Substituting the Roots**:
Substitute \( \alpha = 3\beta \) into the sum of the roots:
\[ 3\beta + \beta = -\frac{b}{a} \]
This simplifies to:
\[ 4\beta = -\frac{b}{a} \quad \text{(Equation 1)} \]

4. **Substituting for Product of Roots**:
Now substitute \( \alpha = 3\beta \) into the product of the roots:
\[ (3\beta)(\beta) = \frac{c}{a} \]
This simplifies to:
\[ 3\beta^2 = \frac{c}{a} \quad \text{(Equation 2)} \]

5. **Expressing \( \beta^2 \)**:
From Equation 1, we can express \( \beta \):
\[ \beta = -\frac{b}{4a} \]
Now, substituting this into Equation 2:
\[ 3\left(-\frac{b}{4a}\right)^2 = \frac{c}{a} \]
Simplifying this gives:
\[ 3\left(\frac{b^2}{16a^2}\right) = \frac{c}{a} \]
Multiplying both sides by \( 16a^2 \):
\[ 3b^2 = 16ac \]

6. **Final Result**:
Rearranging gives us:
\[ 16ac = 3b^2 \]

### Conclusion:
The relationship derived from the conditions given in the problem is:
\[ 16ac = 3b^2 \]
This matches with option B if it is provided in the options.