Introduction:

Complex quadratic equations are a type of quadratic equation whose solutions are complex numbers, typically involving the imaginary unit 'i'. A quadratic equation is of the form ax^2 + bx + c = 0, where a, b, and c are constants.



Understanding Complex Quadratics:



A complex solution arises when the discriminant (b² - 4ac) of the quadratic equation is negative.

This leads to the square root of a negative number, which is not possible in real numbers, hence the solutions are complex.

Finding Solutions to Complex Quadratics:



Use the quadratic formula: x = (-b ± sqrt(b² - 4ac)) / 2a.

If b² - 4ac is negative, the square root part becomes an imaginary number.

The solutions will have both real and imaginary parts.

Example:



Consider the quadratic equation x² + 4x + 5 = 0.

The discriminant is 4² - 4(1)(5) = 16 - 20 = -4, which is negative.

Apply the quadratic formula: x = (-4 ± sqrt(-4)) / 2.

Simplify: x = (-4 ± 2i√6) / 2.

The solutions are: x = -2 + i√6 and x = -2 - i√6.

Key Points to Remember:



Complex solutions occur in conjugate pairs (a + bi and a - bi).

'i' represents the imaginary unit, where i² = -1.