What is the Quadratic Formula?
A quadratic equation is a second-degree polynomial equation in a single variable, standardized as $ax^2 + bx + c = 0$. Because it is a second-degree equation (due to the squared variable), it forms a U-shaped curve called a parabola on a graph, and it will always have two solutions (called roots or x-intercepts).
The mathematical formula used to find these exact roots is:
Understanding the Discriminant (Δ)
The part of the formula under the square root ($b^2 - 4ac$) is called the Discriminant. It is the most important part of the equation because it tells you exactly what kind of roots you are about to find before you even finish the math.
Positive (Δ > 0)
If the discriminant is positive, the equation has two distinct real roots. Visually, this means the parabola crosses the x-axis at two completely different points.
Zero (Δ = 0)
If the discriminant is exactly zero, the equation has one repeated real root. The bottom tip (the vertex) of the parabola perfectly rests on the x-axis.
Negative (Δ < 0)
If the discriminant is negative, you cannot find a real square root. The equation has two complex/imaginary roots. The parabola is floating above or below the x-axis and never touches it.