If z and z̄ are roots of a quadratic with real coefficients, then sum z + z̄ equals:

We have z + z̄ = (a + ib) + (a - ib). That is z + z̄ = 2a. So z + z̄ = 2 Re(z). Complex conjugate pairs are roots of real-coefficient polynomials. Sum and product of conjugates are both real.