ADO - Quadratic Inequalities (Advanced Application)

Quadratic inequalities involve expressions where a quadratic polynomial (degree 2) is compared using inequality signs such as greater than (>), less than (<), greater than or equal to (≥), or less than or equal to (≤). These problems require finding the range of values of the variable that satisfy the inequality.

1. Basic Form

A quadratic inequality is generally written as:

ax^2 + bx + c ; {>, <, \ge, \le} ; 0

Where:

  • a, b, and c are constants

  • a ≠ 0 (otherwise it becomes linear)

2. Steps to Solve Quadratic Inequalities

Step 1: Bring all terms to one side so that the inequality is compared with zero.

Step 2: Factorize the quadratic expression, if possible. If factorization is not straightforward, use the quadratic formula to find the roots.

Step 3: Find the critical points (roots) where the expression becomes zero.

Step 4: Divide the number line into intervals based on these roots.

Step 5: Test a value from each interval to determine whether the expression is positive or negative in that region.

Step 6: Select intervals that satisfy the given inequality.

3. Understanding the Sign of Quadratic Expressions

A quadratic expression represents a parabola. Its sign depends on:

  • The leading coefficient (a)

  • The position of the roots

If a > 0:

  • The parabola opens upward

  • The expression is positive outside the roots and negative between the roots

If a < 0:

  • The parabola opens downward

  • The expression is negative outside the roots and positive between the roots

4. Example

Solve:
x² - 5x + 6 > 0

Factorize:
x² - 5x + 6 = (x - 2)(x - 3)

Critical points:
x = 2 and x = 3

Check intervals:

  • For x < 2 → expression is positive

  • Between 2 and 3 → expression is negative

  • For x > 3 → expression is positive

Final answer:
x < 2 or x > 3

5. Special Cases

  1. Repeated Roots (Discriminant = 0)
    Example: x² - 4x + 4 ≥ 0
    Factor: (x - 2)² ≥ 0
    This expression is always non-negative for all real values of x.

  2. No Real Roots (Discriminant < 0)
    Example: x² + x + 1 > 0
    Since it never touches the x-axis and opens upward, it is always positive.
    So the solution is all real numbers.

6. Graphical Interpretation

Quadratic inequalities can also be solved by sketching the graph of the quadratic equation:

  • Identify where the graph lies above or below the x-axis

  • Above x-axis means positive values

  • Below x-axis means negative values

7. Common Mistakes to Avoid

  • Ignoring the inequality sign while choosing intervals

  • Forgetting to include equality points in ≥ or ≤ cases

  • Not checking the sign of the leading coefficient

  • Incorrect factorization

  • Skipping interval testing

8. Applications in Exams

Quadratic inequalities are frequently used in:

  • Data sufficiency questions

  • Case-based quantitative problems

  • Optimization and constraint-based problems

  • Graph interpretation questions

They test both algebraic understanding and logical reasoning, making them an important topic for competitive exams like ADO.