What is the quadratic formula?

The quadratic formula x = (-b +/- sqrt(b^2 - 4ac)) / (2a) gives the roots (solutions) of any quadratic equation ax^2 + bx + c = 0. The discriminant (b^2 - 4ac) determines whether there are two real roots (positive discriminant), one repeated root (zero discriminant), or no real roots (negative discriminant).

What is vertex form of a quadratic?

Vertex form is y = a(x - h)^2 + k, where (h, k) is the vertex (highest or lowest point) of the parabola. You can convert from standard form by completing the square, or by using h = -b/(2a) and k = f(h). The vertex form makes it easy to read the vertex coordinates and understand transformations.

How does the discriminant affect the graph?

The discriminant (delta = b^2 - 4ac) determines how the parabola intersects the x-axis. When delta > 0, the parabola crosses the x-axis at two points (two real roots). When delta = 0, the vertex touches the x-axis (one repeated root). When delta < 0, the parabola doesn't cross the x-axis at all (no real roots, only complex roots).

Quadratic Equation Explorer

Drag the sliders for a, b, and c. Watch the parabola reshape, roots move, and the vertex shift in real-time.

y = ax² + bx + c

a = 1

1.0

b = 0

0.0

c = -4

-4.0

Properties

Vertex	—
Δ	—
Roots	—
y-int	—
Vertex form	—

The Math Behind It