Forming quadratic equations from their roots
Answers are for parents and teachers.
1
The roots of x² + (k − 2)x + 8 = 0 are r and 2r. Find k.
Product 2r² = 8, so r = 2 or r = −2. Sum 3r = −(k − 2). For r = 2: 6 = 2 − k, k = −4 (x² − 6x + 8, roots 2 and 4). For r = −2: −6 = 2 − k, k = 8 (x² + 6x + 8, roots −2 and −4).
2
−1 + 3i is a root of x² + bx + c = 0 with b and c real. Find b and c.
The other root is −1 − 3i. Sum −2 = −b, so b = 2. Product 1 + 9 = 10 = c.
3
Form the equations with roots (a) 3 + √5 and 3 − √5 (b) 2/3 and −3/2.
(a) sum 6, product 9 − 5 = 4: x² − 6x + 4 = 0. (b) sum −5/6, product −1: x² + (5/6)x − 1 = 0, i.e. 6x² + 5x − 6 = 0.
4
L and M are the roots of x² − 3x − 7 = 0. Form the equation with roots L² and M²; do not solve.
L + M = 3, LM = −7. New sum = 9 + 14 = 23, new product = 49. The equation is x² − 23x + 49 = 0.
5
For 2x² − 6x + 3 = 0 find L² + M² and (L − M)² without solving.
L + M = 3, LM = 3/2. L² + M² = 9 − 3 = 6. (L − M)² = 9 − 6 = 3.