Quadratic Equation Solver (Visual)
Solve quadratic equations (ax² + bx + c = 0) with step-by-step solutions, visual parabola graph, vertex form conversion, discriminant analysis, and root classification. Interactive SVG graph shows roots, vertex, axis of symmetry, and y-intercept.
Solving
x² - 5x + 6 = 0
ax² + bx + c = 0
a (x² coefficient)
b (x coefficient)
c (constant)
Solutions
Discriminant
1
Vertex
(2.5, -0.25)
Axis of Symmetry
x = 2.5
Y-Intercept
(0, 6)
Vertex Form
y = 1(x − 2.5)² − 0.25
Opens upward — vertex is a minimum
Parabola Graph
Step-by-Step Solution
Step 1: Identify coefficients
a = 1, b = -5, c = 6
Step 2: Calculate the discriminant
Δ = b² − 4ac = (-5)² − 4(1)(6)
Δ = 25 − 24 = 1
Step 3: Determine nature of roots
Δ = 1 > 0 → Two distinct real roots
Step 4: Apply the quadratic formula
x = (−b ± √Δ) / 2a
x = (−(-5) ± √1) / (2 × 1)
x = (5 ± 1) / 2
x₁ = 3
x₂ = 2
Step 5: Find vertex & convert to vertex form
h = −b / 2a = −(-5) / (2 × 1) = 2.5
k = f(h) = 1(2.5)² + -5(2.5) + 6 = -0.25
Vertex form: y = 1(x − 2.5)² − 0.25
Quadratic Formula Reference
x = (−b ± √(b² − 4ac)) / 2a
- Discriminant (Δ) > 0 → Two distinct real roots
- Discriminant (Δ) = 0 → One repeated (double) root
- Discriminant (Δ) < 0 → Two complex conjugate roots
- Vertex form: y = a(x − h)² + k, where vertex is (h, k)
- Axis of symmetry: x = −b / 2a
- Sum of roots: x₁ + x₂ = −b/a
- Product of roots: x₁ · x₂ = c/a
How to Use
- Enter your value in the input field
- Click the Calculate/Convert button
- Copy the result to your clipboard
Frequently Asked Questions
- What is the quadratic formula and how does this solver use it?
- The quadratic formula x = (−b ± √(b² − 4ac)) / 2a solves any equation of the form ax² + bx + c = 0. This solver applies the formula step by step, showing the discriminant calculation, root evaluation, and final answers. It handles real and complex roots automatically.
- What does the discriminant tell me about the roots?
- The discriminant Δ = b² − 4ac determines the nature of the roots. If Δ > 0, there are two distinct real roots (parabola crosses x-axis twice). If Δ = 0, there is one repeated root (parabola touches x-axis at one point). If Δ < 0, there are two complex conjugate roots (parabola does not cross the x-axis).
- What is vertex form and why is it useful?
- Vertex form is y = a(x − h)² + k, where (h, k) is the vertex (turning point) of the parabola. It is useful because you can immediately read the vertex coordinates, determine whether the parabola opens up (a > 0) or down (a < 0), and identify the minimum or maximum value of the function (k).
- How do I read the parabola graph?
- The SVG graph shows the parabola curve (blue), roots where it crosses the x-axis (red dots), the vertex or turning point (green dot), the y-intercept (orange dot), and the axis of symmetry (purple dashed line). For complex roots, no red dots appear since the parabola does not intersect the x-axis.
- Can this solver handle complex (imaginary) roots?
- Yes. When the discriminant is negative, the solver calculates complex roots in the form a + bi and a − bi, where i = √(−1). Complex roots always come in conjugate pairs. The step-by-step solution shows how the imaginary part is derived from √(−Δ) / 2a.
- What is the difference between this and the basic quadratic calculator?
- This visual solver adds a step-by-step worked solution, an interactive SVG parabola graph showing roots, vertex, axis of symmetry, and y-intercept, explicit vertex form conversion, and a nature-of-roots indicator badge. The basic calculator provides quick numerical results.