The condition that the roots of the equation `ax^(2)+bx+c=0` be such that one root is n times the other

Options

A. `na^(2) =bc (n+1)^(2)`
B. `nb^(2)=ca(n+1)^(2)`
C. `nc^(2)=ab (n+1)^(2)`
D. none

Answer

To find the condition that the roots of the equation \( ax^2 + bx + c = 0 \) be such that one root is \( n \) times the other, we can follow these steps:

### Step 1: Define the Roots
Let the roots of the quadratic equation be \( \alpha \) and \( n\alpha \), where \( \alpha \) is one root and \( n\alpha \) is the other root.

### Step 2: Use the Sum of Roots Formula
According to Vieta's formulas, the sum of the roots is given by:
\[ \alpha + n\alpha = -\frac{b}{a} \]
This simplifies to:
\[ \alpha(1 + n) = -\frac{b}{a} \quad \text{(1)} \]

### Step 3: Use the Product of Roots Formula
The product of the roots is given by:
\[ \alpha \cdot n\alpha = \frac{c}{a} \]
This simplifies to:
\[ n\alpha^2 = \frac{c}{a} \quad \text{(2)} \]

### Step 4: Eliminate \( \alpha \)
From equation (1), we can express \( \alpha \):
\[ \alpha = -\frac{b}{a(1+n)} \]
Now, substitute this expression for \( \alpha \) into equation (2):
\[ n\left(-\frac{b}{a(1+n)}\right)^2 = \frac{c}{a} \]

### Step 5: Simplify the Equation
Expanding the left-hand side:
\[ n \cdot \frac{b^2}{a^2(1+n)^2} = \frac{c}{a} \]
Now, multiply both sides by \( a^2(1+n)^2 \):
\[ n b^2 = c a (1+n)^2 \]

### Step 6: Final Condition
Thus, we arrive at the condition:
\[ n b^2 = c a (1+n)^2 \]

### Summary
The condition that the roots of the equation \( ax^2 + bx + c = 0 \) be such that one root is \( n \) times the other is given by:
\[ n b^2 = c a (1+n)^2 \]