Q5. Solve √2 x2 + x + √2 = 0.
D = 1 − 4 × √2 × √2 = 1 − 8 = −7, so x = (−1 ± i√7)/(2√2).
Q6. One root of x2 + px + q = 0 (with p, q real) is 2 − 3i. Find p and q.
The other root is the conjugate 2 + 3i. Sum = 4 = −p and product = 4 + 9 = 13 = q, so p = −4 and q = 13.
Q7. If α and β are the roots of x2 + x + 1 = 0, find α2 + β2.
α + β = −1 and αβ = 1, so α2 + β2 = 1 − 2 = −1. (The roots are not real, and the sum of their squares is a negative real number.)
Q8. Show that the roots of x2 + kx − 1 = 0 are real and different for every real k.
D = k2 + 4 > 0 for every real k, so the roots are always real and different.
Q9. Find the quadratic equation whose roots are the squares of the roots of x2 − 3x + 1 = 0.
With α + β = 3 and αβ = 1: α2 + β2 = 9 − 2 = 7 and α2β2 = 1. So x2 − 7x + 1 = 0.