Quadratic Equation Solver

What is a Quadratic Equation?

A quadratic equation is in the form ax² + bx + c = 0, where:

a is the coefficient of the quadratic term (x²) and cannot be zero
b is the coefficient of the linear term (x)
c is the constant term

This is one of the fundamental skills in mathematics, used in fields ranging from physics to economics.

The Discriminant

The discriminant (Δ) is the key number in solving quadratic equations. It is calculated with the formula:

Δ = b^{2} - 4 a c

The value of the discriminant tells us what kind of solutions the equation has:

Δ > 0: Two distinct real solutions
Δ = 0: One real solution (repeated root)
Δ < 0: Two complex solutions (no real solutions)

Solution Formula

The solutions to a quadratic equation are found using the quadratic formula:

x = \frac{- b \pm b ^{2} - 4 a c}{2 a}

When the discriminant is positive, the ± symbol produces two different values:

x_{1} = \frac{- b + Δ}{2 a}, x_{2} = \frac{- b - Δ}{2 a}

Vertex of the Parabola

The quadratic function y = ax² + bx + c forms a parabola. The vertex is the turning point of the parabola, located at:

Vertex = (- \frac{b}{2 a}, - \frac{Δ}{4 a})

When a > 0, the parabola opens upward and the vertex is a minimum point. When a < 0, the parabola opens downward and the vertex is a maximum point.

Example

Let's solve the equation x² - 5x + 6 = 0

Here a = 1, b = -5, c = 6.

Discriminant:
$Δ = (- 5)^{2} - 4 (1) (6) = 25 - 24 = 1$
Solutions:
$x_{1} = \frac{5 + 1}{2} = \frac{5 + 1}{2} = 3$ $x_{2} = \frac{5 - 1}{2} = \frac{5 - 1}{2} = 2$
Vertex:
$x = - \frac{- 5}{2 ( 1 )} = 2.5$ $y = - \frac{1}{4 ( 1 )} = - 0.25$
The vertex is at (2.5, -0.25)

How to Use

Enter the equation coefficients a, b, and c in the respective fields
The calculator automatically displays the discriminant value
Solutions are presented based on the discriminant value:
- Two real solutions (Δ > 0)
- One real solution (Δ = 0)
- Two complex solutions (Δ < 0)
The vertex of the parabola is also shown

Complex Solutions

When the discriminant is negative, the solutions are complex numbers in the form a + bi, where i = √(-1). These solutions always appear in conjugate pairs (a + bi and a - bi).

Interpretation