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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 does the solution x ≤ -7 or x ≥ 16 represent in a quadratic inequality?

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This solution indicates that the values of x that satisfy the inequality are either less than or equal to -7 or greater than or equal to 16.

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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.

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What is the significance of the vertex in a quadratic function?

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The vertex of a quadratic function is the highest or lowest point on the graph, depending on whether it opens upwards or downwards. It helps determine the range of the function.

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What is the discriminant in a quadratic equation?

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The discriminant is the part of the quadratic formula under the square root (b² - 4ac) that determines the nature of the roots. If it's positive, there are two real roots; if zero, one real root; if negative, no real roots.

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How do you graph a quadratic inequality?

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To graph a quadratic inequality, first graph the corresponding quadratic equation as a parabola. Then, use a dashed line for < or > and a solid line for ≤ or ≥. Shade the appropriate region based on the inequality.

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What does the solution -3 ≤ x ≤ 3 indicate in a quadratic inequality?

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This solution indicates that the values of x that satisfy the inequality are between -3 and 3, inclusive.

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

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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 the solution -3 < x < 5.5 represent in a quadratic inequality?

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This solution indicates that the values of x that satisfy the inequality are greater than -3 and less than 5.5.