Quadratic equations: the discriminant and the sum and product of the roots
Answers are for parents and teachers.
1
Find the discriminant and the kind of roots: (a) 6x² − x − 2 = 0 (b) 4x² + 4x + 5 = 0 (c) x(x − 6) = −9.
(a) 1 + 48 = 49 > 0, two different real roots. (b) 16 − 80 = −64 < 0, two complex roots. (c) x² − 6x + 9 = 0, discriminant 0, one repeated root x = 3.
2
Find k if x² − (k + 2)x + 9 = 0 has equal roots.
(k + 2)² − 36 = 0, so k + 2 = 6 or −6: k = 4 or k = −8.
3
Solve 2x² − 3x + 5 = 0 with the general formula.
b² − 4ac = 9 − 40 = −31, so x = (3 ± √31 i)/4.
4
Without solving, find the sum and product of the roots of 7x² − 14x + 5 = 0 and of 2x² = 3x + 9.
First: sum 2, product 5/7. Second: 2x² − 3x − 9 = 0, sum 3/2, product −9/2 (the roots are 3 and −3/2).
5
Show that x² + kx − 4 = 0 has two different real roots for every real k.
b² − 4ac = k² + 16 ≥ 16 > 0 for every k.