If sum of squares of the root of the equation `x^(2) + kx - b = 0` is 2b. Then what is the k equal to ?

Options

A. 1
B. b
C. `-b`
D. 0

Answer

To solve the problem, we need to find the value of \( k \) given that the sum of the squares of the roots of the equation \( x^2 + kx - b = 0 \) is equal to \( 2b \).

### Step-by-Step Solution:

1. **Identify the roots**: Let the roots of the equation \( x^2 + kx - b = 0 \) be \( \alpha \) and \( \beta \).

2. **Use Vieta's Formulas**:
- The sum of the roots \( \alpha + \beta = -\frac{k}{1} = -k \).
- The product of the roots \( \alpha \beta = \frac{-b}{1} = -b \).

3. **Sum of squares of the roots**: The sum of the squares of the roots can be expressed using the formula:
\[ \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta \]
Substituting the values from Vieta's formulas:
\[ \alpha^2 + \beta^2 = (-k)^2 - 2(-b) = k^2 + 2b \]

4. **Set up the equation**: According to the problem, we know that:
\[ \alpha^2 + \beta^2 = 2b \]
Therefore, we can set up the equation:
\[ k^2 + 2b = 2b \]

5. **Simplify the equation**: By subtracting \( 2b \) from both sides, we get:
\[ k^2 = 0 \]

6. **Solve for \( k \)**: Taking the square root of both sides gives:
\[ k = 0 \]

### Final Answer:
Thus, the value of \( k \) is \( 0 \).

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