Quadratic Inequalities

Front

What is a quadratic inequality?

Back

A quadratic inequality is an inequality that involves a quadratic expression, typically in the form ax² + bx + c < 0, ax² + bx + c > 0, ax² + bx + c ≤ 0, or ax² + bx + c ≥ 0.

Front

How do you solve a quadratic inequality?

Back

To solve a quadratic inequality, first solve the corresponding quadratic equation to find the critical points. Then, test intervals between these points to determine where the inequality holds true.

Front

What are critical points in the context of quadratic inequalities?

Back

Critical points are the values of x where the quadratic expression equals zero. They are found by solving the equation ax² + bx + c = 0.

Front

What does the symbol '≥' mean in a quadratic inequality?

Back

The symbol '≥' means 'greater than or equal to', indicating that the values satisfying the inequality can be equal to the critical points.

Front

What does the symbol '<' mean in a quadratic inequality?

Back

The symbol '<' means 'less than', indicating that the values satisfying the inequality cannot include the critical points.

Front

What is the significance of the parabola in quadratic inequalities?

Back

The graph of a quadratic inequality is represented by a parabola, which opens upwards or downwards depending on the sign of the leading coefficient (a). The regions above or below the x-axis indicate where the inequality holds.

Front

How do you determine the intervals for a quadratic inequality?

Back

After finding the critical points, test values from each interval created by these points in the original inequality to see if they satisfy the inequality.

Front

What is the solution set for the inequality 2x² + 5x ≥ 12?

Back

The solution set is (−∞, -4] U [1.5, ∞).

Front

What is the solution set for the inequality 3x - x² > 0?

Back

The solution set is (0, 3).

Front

What is the solution set for the inequality 5x - 10x² < 0?

Back

The solution set is (−∞, 0) U (½, ∞).