If one of the roots of the an equation, `x^(2) - 2x + c =0` is thrice the other, then `c =?`

Options

A. `1/2`
B. `4/3`
C. `-1/2`
D. `3/4`

Explanation

To find the value of \( c \) in the equation \( x^2 - 2x + c = 0 \) given that one root is thrice the other, we can follow these steps:

### Step 1: Define the roots
Let one root be \( \alpha \) and the other root be \( 3\alpha \) (since one root is thrice the other).

### Step 2: Use the sum of the roots
According to Vieta's formulas, the sum of the roots of the quadratic equation \( ax^2 + bx + c = 0 \) is given by:
\[ \text{Sum of roots} = -\frac{b}{a} \]
For our equation \( x^2 - 2x + c = 0 \):
- \( a = 1 \)
- \( b = -2 \)

Thus, the sum of the roots is:
\[ \alpha + 3\alpha = -\frac{-2}{1} = 2 \]
This simplifies to:
\[ 4\alpha = 2 \]

### Step 3: Solve for \( \alpha \)
Now, we can solve for \( \alpha \):
\[ \alpha = \frac{2}{4} = \frac{1}{2} \]

### Step 4: Use the product of the roots
Next, we use the product of the roots, which is given by:
\[ \text{Product of roots} = \frac{c}{a} \]
For our equation:
\[ \alpha \cdot 3\alpha = \frac{c}{1} \]
Substituting \( \alpha \):
\[ \left(\frac{1}{2}\right) \cdot 3\left(\frac{1}{2}\right) = c \]
This simplifies to:
\[ \frac{3}{4} = c \]

### Final Answer
Thus, the value of \( c \) is:
\[ \boxed{\frac{3}{4}} \]

---