Solving 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 does the solution set of a quadratic inequality represent?

Back

The solution set represents the values of x for which the quadratic expression is either less than, greater than, less than or equal to, or greater than or equal to zero.

Front

What is the significance of the critical points in solving quadratic inequalities?

Back

Critical points are the x-values where the quadratic expression equals zero. They divide the number line into intervals that can be tested to find where the inequality holds.

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What does it mean if a quadratic inequality has no solution?

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If a quadratic inequality has no solution, it means that there are no values of x that satisfy the inequality, often occurring when the quadratic opens upwards and the inequality is less than zero.

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What is the graphical representation of a quadratic inequality?

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The graphical representation of a quadratic inequality is a parabola, with the solution set indicated by shading the region above or below the parabola, depending on the inequality.

Front

What is the difference between strict and non-strict inequalities?

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Strict inequalities (<, >) do not include the boundary points, while non-strict inequalities (≤, ≥) include the boundary points in the solution set.

Front

How do you determine the intervals for a quadratic inequality?

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To determine the intervals, find the critical points by solving the quadratic equation, then test values from each interval to see if they satisfy the inequality.

Front

What is the role of the leading coefficient in a quadratic inequality?

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The leading coefficient (a) determines the direction of the parabola: if a > 0, the parabola opens upwards; if a < 0, it opens downwards, affecting the solution set.

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What does it mean if a quadratic inequality has all real numbers as a solution?

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If a quadratic inequality has all real numbers as a solution, it means the quadratic expression is always true for any value of x, often occurring when the quadratic is always above or below the x-axis.