Quadratic Equation Calculator
Solve equations of the form \(ax^2 + bx + c = 0\) with discriminant, roots, and parabola vertex.
x² +
x +
= 0
Roots (\(x_1, x_2\))
x₁ = 3, x₂ = 2
Two distinct real roots
Discriminant (\(\Delta = b^2 - 4ac\))
Δ = 1
Positive (\(\Delta > 0\))
Parabola Vertex \((h, k)\)
(2.5, -0.25)
Opens upward (minimum point)
Axis of Symmetry
x = 2.5
Vertical line through vertex
The Quadratic Formula & How to Calculate
View All Math Formulas Handbook →The roots of any standard quadratic equation are determined using the quadratic formula:
$$ ax^2 + bx + c = 0 \implies x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$
How to Calculate Step-by-Step
Step 1: Compute the Discriminant ($\Delta$)
$$ \Delta = b^2 - 4ac $$
- $\Delta > 0$: Two distinct real roots: $$ x_1 = \frac{-b + \sqrt{\Delta}}{2a}, \quad x_2 = \frac{-b - \sqrt{\Delta}}{2a} $$
- $\Delta = 0$: One repeated double root: $$ x = -\frac{b}{2a} $$
- $\Delta < 0$: Two complex conjugate roots: $$ x = -\frac{b}{2a} \pm \frac{\sqrt{-\Delta}}{2a}\,i $$
Step 2: Find Vertex & Axis of Symmetry
$$ x_v = -\frac{b}{2a}, \quad y_v = c - \frac{b^2}{4a} $$
The axis of symmetry is the vertical line $x = -b / (2a)$.