Complex Numbers and Quadratic Equations - Quadratic Equations with real coefficients

A quadratic equation with real coefficients is expressed in the standard form $ax^2 + bx + c = 0$, where $a, b, c \in \mathbb{R}$ and $a \neq 0$. Visually, this equation represents a parabola that opens upwards if $a > 0$ and downwards if $a < 0$.

The Discriminant, denoted as $D = b^2 - 4ac$, determines the nature of the roots. When $D < 0$, the parabola does not intersect or touch the x-axis at any point, indicating that the roots are not real but complex numbers.

For a quadratic equation with real coefficients, if the discriminant $D$ is negative, the roots are complex and always occur in conjugate pairs. This means if one root is $z = p + iq$, then the other root must be $\bar{z} = p - iq$.

The Fundamental Theorem of Algebra states that a polynomial equation of degree $n$ has exactly $n$ roots in the complex number system. For quadratic equations ($n=2$), there are exactly two roots, which may be real and distinct, real and equal, or complex conjugates.

When solving $ax^2 + bx + c = 0$ with $D < 0$, we introduce the imaginary unit $i = \sqrt{-1}$. The square root of the negative discriminant $\sqrt{D}$ is expressed as $i\sqrt{|D|}$ or $i\sqrt{4ac - b^2}$, allowing us to find solutions in the complex plane.

Graphically, the complex roots of $ax^2 + bx + c = 0$ can be visualized as points in the Argand plane. Because they are conjugates ($p \pm iq$), they are reflections of each other across the real (horizontal) axis.