Quadratic inequalities may seem daunting at first, but with a clear understanding of the underlying concepts and a step-by-step approach, they can be easily solved. In this article, we will discuss how to solve quadratic inequalities to find the possible solutions. Let’s get started!

What are Quadratic Inequalities?

A quadratic inequality is an inequality that contains a quadratic expression. It generally takes the form ax² + bx + c {<, ≤, >, ≥} 0, where a, b, and c are real numbers, and the symbol {<, ≤, >, ≥} represents less than, less than or equal to, greater than, and greater than or equal to, respectively.

Step-by-Step Approach to Solve Quadratic Inequalities

To solve a quadratic inequality, follow these steps:

  • Rearrange the inequality so that one side of the equation is equal to zero. For example, if you have 5x² – 3x – 2 > 0, rewrite it as 5x² – 3x – 2 = 0.
  • Factorize the quadratic expression if possible. Using the quadratic formula x = (-b ± √(b²-4ac)) / (2a) can help determine the roots.
  • Identify the critical points, which are the values of x that make the quadratic expression equal to zero. These points divide the number line into intervals.
  • Test a point from each interval in the original inequality. Choose a point that lies within the interval but not on a critical point.
  • Determine the sign of the expression for each interval based on the tests. If the expression is positive, use the corresponding inequality symbol from the original inequality. If the expression is negative, use the opposite inequality symbol.
  • Write down the solution by combining the inequalities for all intervals.

Example

Let’s solve the quadratic inequality x² – 3x – 4 > 0 using the steps mentioned above:

  • Rearrange the inequality: x² – 3x – 4 = 0
  • Factorize the expression: (x – 4)(x + 1) = 0
  • Critical points: x = 4 and x = -1
  • Choose a test point from each interval: x = -2 and x = 0
  • Test the intervals: For x < -1, (-2)² – 3(-2) – 4 = 6 > 0. For -1 < x < 4, 0² – 3(0) – 4 = -4 < 0. For x > 4, (0)² – 3(0) – 4 = -4 < 0.
  • Solution: The inequality is satisfied when x < -1 or x > 4.

By following these steps with any quadratic inequality, you can easily find the solution set. Practice and familiarity with quadratic equations will help you become more efficient in solving these inequalities. Happy problem-solving!

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