TOPIC: Quadratic Equations – Advanced Problems
1. Solve the equation:
3x² – 7x – 6 = 0
Solution:
Using quadratic formula:
x = [7 ± √(49 + 72)] / 6
x = [7 ± √121] / 6
x = (7 + 11)/6 , (7 – 11)/6
x = 3 , x = –2/3
2. If α and β are roots of the equation:
2x² – 5x + 3 = 0
Find (α² + β²).
Solution:
α + β = 5/2
αβ = 3/2
α² + β² = (α + β)² – 2αβ
= (25/4) – 3
= 13/4
3. Find the value of k for which the equation:
x² + kx + 9 = 0
has equal roots.
Solution:
For equal roots:
b² – 4ac = 0
k² – 36 = 0
k = ±6
