Master solving quadratic inequalities algebraically and graphically, and learn to find maximum and minimum values of quadratic functions for real-world optimization problems!
Solve the quadratic inequality!
Inequality:
Find the maximum value!
Function:
A quadratic inequality is an inequality involving a quadratic expression. Examples include x² - 4x + 3 > 0, 2x² - 8 ≤ 0, etc.
Key Steps to Solve:
Step 1: Factor: (x - 1)(x - 3) > 0
Step 2: Critical points: x = 1, x = 3
Step 3: Test intervals:
• x < 1: Test x = 0: (0-1)(0-3) = 3 > 0 ✓
• 1 < x < 3: Test x = 2: (2-1)(2-3) = -1 < 0 ✗
• x > 3: Test x = 4: (4-1)(4-3) = 3 > 0 ✓
Solution: x < 1 or x > 3
Step 1: Factor: (x - 3)(x + 3) ≤ 0
Step 2: Critical points: x = -3, x = 3
Step 3: Since parabola opens upward and we want ≤ 0:
The function is negative between the roots
Solution: -3 ≤ x ≤ 3