Understanding Quadratic Equations and the Quadratic Formula
A **quadratic equation** is a second-order polynomial equation in a single variable. The standard representation is:
ax² + bx + c = 0
To determine the values of x (known as the roots or solutions) where this equation holds true, we use the standard **quadratic formula**:
x = [ -b ± √(b² - 4ac) ] / (2a)
Role of the Discriminant (D)
The value under the square root, D = b² - 4ac, is called the **discriminant**. It acts as a classifier indicating the number and nature of the roots:
- D > 0: Two distinct real numbers as roots. The parabola crosses the x-axis at two separate points.
- D = 0: One real number as a double root. The vertex of the parabola lies exactly on the x-axis.
- D < 0: Two complex conjugate numbers as roots (containing imaginary parts, written with i). The parabola does not intersect the x-axis.
Parabolic Geometry
A quadratic equation graphically translates to a **parabola**. The peak or trough of this curve is its **vertex** (xv, yv). The vertical line passing through this vertex is the **axis of symmetry** (x = -b / 2a), dividing the parabola into symmetrical halves.