Solving a Quadratic Inequality?

📌 CONCEPT: A quadratic inequality is a mathematical statement of the form f(z) ≥ 0, where f(z) is a quadratic function, and we need to find the values of z that satisfy this condition.

📐 RULE / FORMULA: The roots of a quadratic function f(z) = az^2 + bz + c are given by the quadratic formula, but for a quadratic inequality, we need to consider the sign of the quadratic function in different intervals.

💡 WORKED EXAMPLE: Consider the quadratic function f(z) = z^2 + 4z + 8. To find the points where the graph crosses the imaginary axis, we need to find the roots of the equation f(z) = 0. Using the quadratic formula, we get z = (-b ± √(b^2 - 4ac)) / 2a. In this case, a = 1, b = 4, and c = 8. Plugging these values into the quadratic formula, we get z = (-4 ± √(16 - 32)) / 2 = (-4 ± √(-16)) / 2. This gives us two complex roots, z = -2 ± 2i.

⚠️ COMMON MISTAKE: Students often forget to consider the sign of the quadratic function in different intervals, which can lead to incorrect solutions.