If one root of the quadratic equation `ax^(2)+bx+c=0` is double the other then which one of the following is correct?

Options

A. `b^(2)=3ac`
B. `2b^(2)=5ac`
C. `2b^(2)=9ac`
D. `2b^(2) gt 9ac`

Answer

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

### Step 1: Use Vieta's Formulas
According to Vieta's formulas, the sum and product of the roots of the quadratic equation can be expressed as:
- The sum of the roots \( \alpha + \beta = -\frac{b}{a} \)
- The product of the roots \( \alpha \beta = \frac{c}{a} \)

### Step 2: Substitute the Roots
Substituting \( \alpha = 2\beta \) into the sum of the roots:
\[ 2\beta + \beta = -\frac{b}{a} \]
This simplifies to:
\[ 3\beta = -\frac{b}{a} \]
From this, we can express \( \beta \):
\[ \beta = -\frac{b}{3a} \]

### Step 3: Find the Product of the Roots
Now, substituting \( \alpha = 2\beta \) into the product of the roots:
\[ (2\beta) \beta = \frac{c}{a} \]
This simplifies to:
\[ 2\beta^2 = \frac{c}{a} \]

### Step 4: Substitute the Value of \( \beta \)
Now, substitute \( \beta = -\frac{b}{3a} \) into the product equation:
\[ 2\left(-\frac{b}{3a}\right)^2 = \frac{c}{a} \]
Calculating \( \left(-\frac{b}{3a}\right)^2 \):
\[ 2\left(\frac{b^2}{9a^2}\right) = \frac{c}{a} \]
This simplifies to:
\[ \frac{2b^2}{9a^2} = \frac{c}{a} \]

### Step 5: Cross-Multiply
Cross-multiplying gives:
\[ 2b^2 = 9ac \]

### Conclusion
Thus, we find that the correct relationship is:
\[ 2b^2 = 9ac \]
This corresponds to the third option.