If α and β are the roots of x^2 - 30x + 6, what is the value of α^2 + β^2?

If α and β are the roots of

x^2 - 30x + 6,

what is the value of α^2 + β^2?

Vieta’s formulae (Theory of Roots and Equations) state that

if α and β are the roots of the quadratic equation Ax2 + Bx + C,

then α + β = −B/A and α ⋅ β = C/A.

This follows by comparing coefficients of

Ax2 + Bx + C = A(x - α)(x - β) = Ax2 - A(α + β)x + Aα⋅β

α + β = 30

α ⋅ β = 6

(α + β)^2 = 30^2

(α + β)^2 - 2αβ = 900 - 12

α^2 + β^2 = 888

x = (30 +/- sqrt(900 - 24)) / 2

x = (30 +/- sqrt(876)) / 2

x = (30 +/- 2 * sqrt(219)) / 2

x = 15 +/- sqrt(219)

a = 15 + sqrt(219)

b = 15 - sqrt(219)

a^2 + b^2 =>

(15 + sqrt(219))^2 + (15 - sqrt(219))^2 =>

225 + 30 * sqrt(219) + 219 + 225 - 30 * sqrt(219) + 219 =>

450 + 438 =>

888