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

Back

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.

Front

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

Back

If a quadratic inequality has no solution, it means that there are no values of x that satisfy the inequality.

Front

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.

Front

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.

Front

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

Back

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

Front

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.

Front

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.