Quadratic Inequalities

Quadratic Inequalities

Understanding Quadratic Inequalities

• Quadratic inequalities in one variable are inequalities which can be represented in the form ax² + bx + c < 0, ax² + bx + c ≤ 0, ax² + bx + c > 0 or ax² + bx + c ≥ 0, where ‘a’, ‘b’ and ‘c’ are constants.

Solving Quadratic Inequalities

• The first step to solving a quadratic inequality is to rewrite the inequality as an equation and solve that equation.

• The solutions to that equation, also known as the roots or zeros, split the number line into intervals. These intervals represent potential solutions to the inequality.

• Using a sign chart or testing points from each interval in the original inequality helps determine whether the interval is part of the solution set.

• Quadratic inequalities can have two solutions, one solution, or no solutions.

Interpreting Quadratic Inequalities

• Graphing the associated quadratic function can provide a visual understanding of the solutions and make it easier to solve the inequality.

• The graph crosses the x-axis at the solutions to the associated equation, splitting the x-axis into the intervals.

• For ‘greater than’ inequalities (> or ≥), the solution includes values of x where the graph of the quadratic function is above the x-axis.

• For ‘less than’ inequalities (< or ≤), the solution includes values of x where the graph of the quadratic function is below the x-axis.

All of the above steps firmly point to the fact that understanding and correctly applying the principles of quadratic inequalities can lead to successful solutions.