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If the sum of the roots of x^(2) + bx + ...

If the sum of the roots of `x^(2) + bx + 1 = 0`, is equal to the sum of their squares, then b =

A

1, 2

B

1, -2

C

-1, 2

D

-1, -2

Text Solution

AI Generated Solution

The correct Answer is:
To solve the problem, we need to find the value of \( b \) given that the sum of the roots of the quadratic equation \( x^2 + bx + 1 = 0 \) is equal to the sum of their squares. ### Step-by-Step Solution: 1. **Identify the roots and their relationships**: Let the roots of the quadratic equation \( x^2 + bx + 1 = 0 \) be \( \alpha \) and \( \beta \). According to Vieta's formulas: - The sum of the roots \( \alpha + \beta = -b \) - The product of the roots \( \alpha \beta = 1 \) 2. **Express the sum of the squares of the roots**: The sum of the squares of the roots can be expressed in terms of the sum and product of the roots: \[ \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta \] Substituting the values from Vieta's formulas: \[ \alpha^2 + \beta^2 = (-b)^2 - 2 \cdot 1 = b^2 - 2 \] 3. **Set up the equation from the given condition**: We are given that the sum of the roots is equal to the sum of their squares: \[ \alpha + \beta = \alpha^2 + \beta^2 \] Substituting the expressions we found: \[ -b = b^2 - 2 \] 4. **Rearranging the equation**: Rearranging the equation gives: \[ b^2 + b - 2 = 0 \] 5. **Factoring the quadratic equation**: We can factor the quadratic: \[ (b + 2)(b - 1) = 0 \] 6. **Finding the values of \( b \)**: Setting each factor to zero gives us: \[ b + 2 = 0 \quad \Rightarrow \quad b = -2 \] \[ b - 1 = 0 \quad \Rightarrow \quad b = 1 \] Thus, the values of \( b \) that satisfy the condition are \( b = 1 \) and \( b = -2 \). ### Final Answer: The values of \( b \) are \( 1 \) and \( -2 \). ---
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