solving quadratic inequalities

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What is a quadratic inequality?

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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.

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How do you solve a quadratic inequality?

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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.

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What are critical points in the context of quadratic inequalities?

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

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What does the graph of a quadratic function look like?

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The graph of a quadratic function is a parabola, which can open upwards or downwards depending on the sign of the leading coefficient (a).

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

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The leading coefficient determines the direction of the parabola: if it is positive, the parabola opens upwards; if negative, it opens downwards.

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What does it mean for a quadratic expression to be less than zero?

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A quadratic expression being less than zero means that the values of the expression are negative, which corresponds to the regions of the graph that lie below the x-axis.

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What is the solution set of a quadratic inequality?

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The solution set of a quadratic inequality is the range of x-values for which the inequality holds true, often expressed in interval notation.

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How do you test intervals when solving quadratic inequalities?

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To test intervals, choose a test point from each interval created by the critical points and substitute it into the inequality to see if it satisfies the condition.

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What is interval notation?

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Interval notation is a way of writing subsets of the real number line, using parentheses and brackets to indicate whether endpoints are included or excluded.

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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 x-values that satisfy the inequality, often occurring when the parabola does not intersect the x-axis.