If-and-are-the-roots-of-x-2-a-1-x-1-2-a-2-a-1-0-then-2-2-

Question Number 62611 by hovea cw last updated on 23/Jun/19

If α and β are the roots of

x^{2} - (a + 1) x + \frac{1}{2} (a^{2} + a + 1) = 0 then

α^{2} + β^{2} =_____.

Commented by Prithwish sen last updated on 23/Jun/19

α + β = (a + 1)

2 α β = (a^{2} + a + 1)

∴ α^{2} + β^{2} = {(α + β)}^{2} - 2 α β = {(a + 1)}^{2} - (a^{2} + a + 1)

= a

Commented by mathmax by abdo last updated on 24/Jun/19

α^{2} + β^{2} = {(α + β)}^{2} - 2 α β = {(a + 1)}^{2} - 2 \frac{1}{2} (a^{2} + a + 1)

= a^{2} + 2 a + 1 - a^{2} - a - 1 = a .