A real value of a, for which the sum of the roots of the equation `x^(2)-2ax+2a-1=0` is equal to the sum of the square of its roots, is

Options

A. `(1)/(2)`
B. `(3)/(2)`
C. `(5)/(2)`
D. 2

Answer

To solve the problem, we need to find the real value of \( a \) for which the sum of the roots of the equation \( x^2 - 2ax + 2a - 1 = 0 \) is equal to the sum of the squares of its roots.

### Step 1: Identify the coefficients of the quadratic equation
The given quadratic equation is:
\[ x^2 - 2ax + (2a - 1) = 0 \]
Here, the coefficients are:
- \( A = 1 \) (coefficient of \( x^2 \))
- \( B = -2a \) (coefficient of \( x \))
- \( C = 2a - 1 \) (constant term)

### Step 2: Calculate the sum of the roots
The sum of the roots \( \alpha + \beta \) of a quadratic equation \( Ax^2 + Bx + C = 0 \) is given by:
\[ \alpha + \beta = -\frac{B}{A} = -\frac{-2a}{1} = 2a \]

### Step 3: Calculate the sum of the squares of the roots
The sum of the squares of the roots \( \alpha^2 + \beta^2 \) can be calculated using the identity:
\[ \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta \]
We already found \( \alpha + \beta = 2a \). Now, we need to find \( \alpha\beta \):
\[ \alpha\beta = \frac{C}{A} = \frac{2a - 1}{1} = 2a - 1 \]
Now substituting these values into the identity:
\[ \alpha^2 + \beta^2 = (2a)^2 - 2(2a - 1) = 4a^2 - 2(2a - 1) = 4a^2 - 4a + 2 \]

### Step 4: Set the sum of the roots equal to the sum of the squares of the roots
We need to set the two expressions equal to each other:
\[ 2a = 4a^2 - 4a + 2 \]

### Step 5: Rearrange the equation
Rearranging gives:
\[ 4a^2 - 4a - 2a + 2 = 0 \implies 4a^2 - 6a + 2 = 0 \]

### Step 6: Simplify the equation
Dividing the entire equation by 2:
\[ 2a^2 - 3a + 1 = 0 \]

### Step 7: Factor the quadratic equation
Now we will factor the quadratic:
\[ 2a^2 - 2a - a + 1 = 0 \]
This can be factored as:
\[ (2a - 1)(a - 1) = 0 \]

### Step 8: Solve for \( a \)
Setting each factor to zero gives us:
1. \( 2a - 1 = 0 \implies a = \frac{1}{2} \)
2. \( a - 1 = 0 \implies a = 1 \)

### Conclusion
The values of \( a \) that satisfy the condition are \( a = \frac{1}{2} \) and \( a = 1 \).