What is the method to solve quadratic inequalities?

Questions : What is the method to solve quadratic inequalities?

Answer:  

 First, solve the corresponding quadratic equation to find its roots. Then, determine the intervals where the quadratic expression is positive or negative by testing values from each interval. This helps in identifying the solution set for the inequality.

Additional Information

1. Standard Form

• Rewrite the inequality so that one side is zero. This will give you the general form:
  • ax² + bx + c < 0
  • ax² + bx + c ≤ 0
  • ax² + bx + c > 0
  • ax² + bx + c ≥ 0

2. Find the Critical Points

• Treat the inequality as an equation and solve for x. You can use:

  • Factoring
  • Quadratic Formula

• These solutions are your critical points. They divide the number line into intervals.

3. Test Intervals

• Choose a test value within each interval created by the critical points.
• Substitute that test value into the original inequality.
• Determine if the inequality is true or false for that test value.

4. Write the Solution

• Based on the true/false results from your test values, identify the intervals that satisfy the original inequality.
• Write the solution using interval notation.

Example

Let's solve the inequality: x² - x - 6 < 0

1. Standard Form: It's already in standard form.

2. Critical Points:

   • Factor: (x - 3)(x + 2) = 0
   • Solutions: x = 3 and x = -2

3. Test Intervals:

   • Interval 1: x < -2 (e.g., x = -3)
     • (-3)² - (-3) - 6 = 6 > 0 (False)
   • Interval 2: -2 < x < 3 (e.g., x = 0)
     • (0)² - (0) - 6 = -6 < 0 (True)
   • Interval 3: x > 3 (e.g., x = 4)
     • (4)² - (4) - 6 = 6 > 0 (False)

4. Solution:

   • The inequality is true for the interval -2 < x < 3.
   • Interval notation: (-2, 3)

Important Notes

• Endpoints: If the inequality includes "or equal to" (≤ or ≥), the critical points are included in the solution (use brackets [ ] in interval notation). Otherwise, they are not included (use parentheses ( )).
• Graphing: You can also graph the quadratic function to visualize the solution. The intervals where the graph is above or below the x-axis will correspond to the solution of the inequality.

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