Complex Roots and Coefficients?

📌 CONCEPT: When the coefficients of a quadratic equation are multiplied by a complex number, the nature of its roots remains unchanged, but their values are altered accordingly.

📐 RULE / FORMULA: This is based on the property that the roots of a quadratic equation ax^2 + bx + c = 0 are given by the quadratic formula: x = (-b ± √(b^2 - 4ac)) / 2a. If we multiply the coefficients by a complex number k, the roots change to (-kb ± √((kb)^2 - 4(ak)(ck))) / 2(ak).

💡 WORKED EXAMPLE: Consider the quadratic equation x^2 + 2x + 2 = 0. If we multiply its coefficients by 2i, we get (2i)x^2 + (4i)x + 2i = 0. The roots of this new equation can be found using the quadratic formula. Let's calculate the discriminant: (4i)^2 - 4(2i)(2i) = -16 - 16 = -32. Since the discriminant is negative, the roots of the new equation are complex.

⚠️ COMMON MISTAKE: Students often assume that multiplying the coefficients of a quadratic equation by a complex number will change the nature of its roots from real to complex or vice versa. However, the nature of the roots remains the same, and the values change in accordance with the complex number used to multiply the coefficients.