If `alpha` and `beta`are the roots of `x^(2) - (a+1)x + 1/2 (a^(2) + a+1) = 0` then `alpha^(2) + beta^(2) = "_____"`.

Options

A. A and B
B. `a^(2)`
C. 2a
D. 1

Answer

A

Explanation

`x^(2) - (a+1)x + 1/2 (a^(2) + a + 1) = 0`
`alpha + beta = a+1`
`alpha beta = 1/2 (a^(2) + a + 1)`
`alpha^(2) + beta^(2) = (alpha + beta)^(2) - 2alphabeta`
`= (a+1)^(2) - 2[1/2(a^(2) +a+1)]`
`= a^(2) + 2a + 1 - a^(2) - a - 1 = a`.