Exit Pass: 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 it mean when a quadratic inequality has 'no solution'?

Back

A quadratic inequality has 'no solution' when the quadratic expression does not satisfy the inequality for any real number, often occurring when the parabola opens downwards and the vertex is above the x-axis.

Front

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

Back

The vertex of a quadratic function is the highest or lowest point of the parabola, which helps determine the range of values for which the quadratic expression is positive or negative.

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 graph the solution of a quadratic inequality?

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To graph the solution of a quadratic inequality, first plot the critical points on a number line, then shade the regions that satisfy the inequality, using open circles for strict inequalities and closed circles for non-strict inequalities.

Front

What is the role of the discriminant in solving quadratic inequalities?

Back

The discriminant (b² - 4ac) determines the nature of the roots of the quadratic equation. If it is positive, there are two distinct real roots; if zero, one real root; and if negative, no real roots.

Front

How do you determine the intervals for testing in a quadratic inequality?

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After finding the critical points from the quadratic equation, the number line is divided into intervals based on these points. Each interval is then tested to see if it satisfies the inequality.

Front

What does the solution set 'all reals' indicate in a quadratic inequality?

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The solution set 'all reals' indicates that the quadratic expression is either always positive or always negative for all values of x.

Front

What is the importance of rearranging terms in a quadratic inequality?

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Rearranging terms to have the quadratic expression on one side and zero on the other side simplifies the process of finding critical points and testing intervals.