Question

If \( \alpha, \beta \) are the roots of \( 2x^2 - 4x + 5 = 0 \), then \( (\alpha + 1)(\beta + 1) = \):

• A. \( \frac{11}{2} \)
• B. \( 2 \)
• C. \( 1 \)
• D. \( 0 \)

Hint:

For the product of shifted roots, use the relation: \( (\alpha + 1)(\beta + 1) = \alpha\beta + \alpha + \beta + 1 \).

Answer 1

The Correct Option is A

Given the quadratic equation: \[ 2x^2 - 4x + 5 = 0 \] Step 1: Sum and Product of Roots
For a general quadratic equation \( ax^2 + bx + c = 0 \), the sum and product of the roots \( \alpha \) and \( \beta \) are given by: \[ \alpha + \beta = -\frac{b}{a}, \quad \alpha \beta = \frac{c}{a} \] Substituting the coefficients \( a = 2 \), \( b = -4 \), and \( c = 5 \): \[ \alpha + \beta = -\frac{-4}{2} = 2 \] \[ \alpha \beta = \frac{5}{2} \] Step 2: Expand \( (\alpha + 1)(\beta + 1) \)
We need to find the value of \( (\alpha + 1)(\beta + 1) \). Let's expand this expression: \[ (\alpha + 1)(\beta + 1) = \alpha \beta + \alpha + \beta + 1 \] Substituting the known values: \[ \alpha \beta + \alpha + \beta + 1 = \frac{5}{2} + 2 + 1 = \frac{5}{2} + \frac{4}{2} + \frac{2}{2} = \frac{11}{2} \] Conclusion
The value of \( (\alpha + 1)(\beta + 1) \) is \( \frac{11}{2} \), which corresponds to option \boxed{1}.