Solving Quadratic Equations with Complex Roots

Complex numbers are a type of number that can be expressed in the form a + bi, where ‘a’ is the real part and ‘bi’ is the imaginary part. ‘i’ is the square root of -1.
Complex numbers include all real numbers (when b is zero) and all pure imaginary numbers (when a is zero).
When graphing complex numbers, the real part is represented on the x-axis and the imaginary part is represented on the y-axis.

Complex roots of a quadratic equation occur when the discriminant (b^2 - 4ac) is less than zero in the quadratic formula x = [-b ± sqrt(b^2 - 4ac)] / 2a.
Each complex root is a complex conjugate of the other. If x = a + bi is a root, then x = a - bi will be the other root.
The fundamental theorem in algebra indicates that every polynomial equation of degree n has exactly n roots.

A quadratic equation with coefficients a, b, and c respectively and roots x = (-b ± sqrt(b^2 - 4ac)) / 2a will have complex roots when the discriminant is less than 0.
The discriminant, b^2 - 4ac, is the term inside the square root in the quadratic formula. If b^2 - 4ac < 0, then we are taking a square root of a negative number, resulting in a complex number.
For the quadratic equation x^2 + bx + c = 0, when the discriminant is negative, the complex roots can be expressed as x = (-b ± sqrt(b^2 - 4ac))/ (2a), or x = -b/2a ± (sqrt(abs(b^2 - 4ac))/ (2a)i where abs() stands for absolute value.

Complex roots can be represented in two forms: rectangular form and polar form.
Rectangular form: a + bi, shows the component parts.
Polar form: r(cos θ + i sin θ), shows the magnitude (r) and direction (θ) of the complex number.
The polar form is more useful for multiplication, division, and finding power of complex numbers.

Complex roots are used in many fields including engineering, physics, computer science, and signal processing.
Complex numbers are vital in studying oscillations, wave mechanics, and alternating current circuits.