If one root of the equation `ax^2+bx+c=0` is double the other, then the relation between `a,b,c` is

Options

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

Answer

To find the relation between \( a, b, c \) when one root of the equation \( ax^2 + bx + c = 0 \) is double the other, we can follow these steps:

### Step 1: Define the Roots
Let the roots of the equation be \( \alpha \) and \( \beta \). Given that one root is double the other, we can express the roots as:
\[ \beta = 2\alpha \]

### Step 2: Use the Sum of Roots
According to Vieta's formulas, the sum of the roots of the quadratic equation \( ax^2 + bx + c = 0 \) is given by:
\[ \alpha + \beta = -\frac{b}{a} \]
Substituting \( \beta = 2\alpha \) into the equation:
\[ \alpha + 2\alpha = -\frac{b}{a} \]
This simplifies to:
\[ 3\alpha = -\frac{b}{a} \]
From this, we can express \( \alpha \) in terms of \( b \) and \( a \):
\[ \alpha = -\frac{b}{3a} \]

### Step 3: Use the Product of Roots
Again, according to Vieta's formulas, the product of the roots is given by:
\[ \alpha \cdot \beta = \frac{c}{a} \]
Substituting \( \beta = 2\alpha \):
\[ \alpha \cdot 2\alpha = \frac{c}{a} \]
This simplifies to:
\[ 2\alpha^2 = \frac{c}{a} \]

### Step 4: Substitute \( \alpha \) into the Product Equation
Now, substitute the value of \( \alpha \) from Step 2 into the product equation:
\[ 2\left(-\frac{b}{3a}\right)^2 = \frac{c}{a} \]
Calculating \( \left(-\frac{b}{3a}\right)^2 \):
\[ 2 \cdot \frac{b^2}{9a^2} = \frac{c}{a} \]
This leads to:
\[ \frac{2b^2}{9a^2} = \frac{c}{a} \]

### Step 5: Clear the Denominator
To eliminate the fraction, multiply both sides by \( 9a^2 \):
\[ 2b^2 = 9ac \]

### Conclusion
Thus, the relation between \( a, b, c \) is:
\[ 2b^2 = 9ac \]