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 rewrite it in standard form, then find the roots of the corresponding quadratic equation. Use test points in the intervals defined by the roots to determine where the inequality holds true.

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

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It means that there are no values of x that satisfy the inequality, often occurring when the quadratic expression is always positive or always negative.

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What does it mean if a quadratic inequality has 'all reals' as a solution?

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It means that every real number satisfies the inequality, typically when the quadratic expression is always less than or equal to zero or greater than or equal to zero.

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

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The vertex is the highest or lowest point of the parabola represented by the quadratic function, and it helps determine the direction of the inequality.

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What is the standard form of a quadratic equation?

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The standard form of a quadratic equation is ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0.

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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, given by b² - 4ac. It determines the nature of the roots: if positive, there are two distinct 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, determine the regions that satisfy the inequality by testing points or using the vertex.

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What does the solution set '-3 ≤ x ≤ 3' represent?

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It represents all values of x that are between -3 and 3, inclusive.

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What does the solution set '-5 < x < 7' represent?

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It represents all values of x that are greater than -5 and less than 7, not including -5 and 7.