Solving Quadratic Inequalities

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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?

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Critical points are the values of the variable that make the quadratic expression equal to zero. They are found by solving the equation ax² + bx + c = 0.

Front

What does it mean if a quadratic inequality is 'greater than zero'?

Back

If a quadratic inequality is greater than zero (e.g., ax² + bx + c > 0), it means that the quadratic expression is positive in certain intervals of the x-axis.

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What does it mean if a quadratic inequality is 'less than zero'?

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If a quadratic inequality is less than zero (e.g., ax² + bx + c < 0), it means that the quadratic expression is negative in certain intervals of the x-axis.

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What is the significance of the direction of the inequality sign?

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The direction of the inequality sign indicates whether the solution includes the boundary points (≥ or ≤) or excludes them (> or <).

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How can you represent the solution of a quadratic inequality on a number line?

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The solution can be represented on a number line by shading the intervals that satisfy the inequality and using open or closed circles to indicate whether the endpoints are included.

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What is the difference between 'and' and 'or' in the context of quadratic inequalities?

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'And' indicates that both conditions must be true simultaneously (intersection of intervals), while 'or' indicates that at least one condition must be true (union of intervals).

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What is an example of a quadratic inequality?

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An example of a quadratic inequality is x² - 5x + 6 < 0.

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How do you find the intervals where a quadratic inequality holds true?

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After determining the critical points, test values from each interval in the inequality to see if they satisfy the condition.