Find the condition that, for the equation `ax^(2)+bx+c=0` one root is m times the other.

Answer

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

### Step 1: Define the Roots
Let the roots of the equation be \( t \) and \( mt \) (where \( m \) is a constant).

### Step 2: Use the Sum of Roots
According to Vieta's formulas, the sum of the roots is given by:
\[ t + mt = -\frac{b}{a} \]
Factoring out \( t \) from the left side, we have:
\[ t(1 + m) = -\frac{b}{a} \]
From this, we can express \( t \) as:
\[ t = -\frac{b}{a(1 + m)} \]

### Step 3: Use the Product of Roots
The product of the roots is given by:
\[ t \cdot mt = \frac{c}{a} \]
This simplifies to:
\[ mt^2 = \frac{c}{a} \]

### Step 4: Substitute \( t \) into the Product Equation
Now, substitute the value of \( t \) from Step 2 into the product equation:
\[ m \left(-\frac{b}{a(1 + m)}\right)^2 = \frac{c}{a} \]
This expands to:
\[ m \cdot \frac{b^2}{a^2(1 + m)^2} = \frac{c}{a} \]

### Step 5: Clear the Denominator
Multiply both sides by \( a^2(1 + m)^2 \) to eliminate the denominator:
\[ m b^2 = c a (1 + m)^2 \]

### Step 6: Final Condition
Thus, the condition that one root is \( m \) times the other is:
\[ m b^2 = c a (1 + m)^2 \]

### Summary
The condition for the quadratic equation \( ax^2 + bx + c = 0 \) to have one root \( m \) times the other is given by:
\[ m b^2 = c a (1 + m)^2 \]