Scientific Quadratic Equation Solver – Complete Guide with Steps and Graph

Quadratic equations are fundamental in mathematics and appear in various fields such as physics, engineering, finance, and computer science. A quadratic equation has the form:

ax² + bx + c = 0

Where:

• a = coefficient of x² (cannot be 0)
  
• b = coefficient of x
  
• c = constant term
  

A Scientific Quadratic Equation Solver not only calculates the roots but also provides step-by-step solutions and a graphical representation to help you understand the behavior of the quadratic function.

Understanding Quadratic Equations

• Roots (solutions): Values of x that satisfy the equation.
  
• Discriminant (D): Determines the nature of roots.
  
• Vertex: The maximum or minimum point on the parabola.
  

Graph: Visualizes how the quadratic function behaves.

How the Calculator Works – Step by Step

Step 1: Enter Coefficients

• Coefficient a (x²) – Must be non-zero. Example: 1
  
• Coefficient b (x) – Example: -3
  
• Constant c – Example: 2
  

Step 2: Calculate the Discriminant

The discriminant (D) is calculated using:

D = b² – 4ac

Interpretation:

• D > 0 → Two distinct real roots
  
• D = 0 → One repeated real root
  
• D < 0 → Two complex roots
  

Step 3: Solve for Roots

Using the quadratic formula:

Root 1 = (-b + √D) / (2a)  

Root 2 = (-b – √D) / (2a)

• If D < 0, the calculator returns complex roots:
  

Root 1 = (-b + i√|D|) / (2a)  

Root 2 = (-b – i√|D|) / (2a)

Step 4: Calculate the Vertex

The vertex gives the highest or lowest point of the parabola:

x_vertex = -b / (2a)  

y_vertex = a*(x_vertex)² + b*(x_vertex) + c

Step 5: Plot the Graph

The graph shows:

• Parabola shape (upward if a > 0, downward if a < 0)
  
• Intersection with x-axis (roots)
  
• Intersection with y-axis (c)
  

Vertex (maximum or minimum)

Formula Summary

1. Discriminant:

D = b² – 4ac

2. Roots:

x = (-b ± √D) / (2a)

3. Vertex:

x_vertex = -b / 2a  

y_vertex = f(x_vertex) = a*(x_vertex)² + b*(x_vertex) + cGraph: Plot f(x) = ax² + bx + c

Real-Life Examples

Example 1 – Two Distinct Real Roots

Equation: x² – 3x + 2 = 0

• a = 1, b = -3, c = 2
  
• Discriminant: D = (-3)² – 412 = 1
  
• Roots:
  

Root 1 = (3 + 1)/2 = 2  

Root 2 = (3 – 1)/2 = 1

• Vertex:
  

x_vertex = 3/2 = 1.5  

y_vertex = 1*(1.5)² – 3*1.5 + 2 = -0.25

Graph shows parabola crossing x-axis at 1 and 2, vertex at (1.5, -0.25).

Example 2 – Repeated Root

Equation: x² – 4x + 4 = 0

• D = (-4)² – 414 = 0
  
• Roots: x = 2 (repeated)
  
• Vertex: (2, 0)
  

Example 3 – Complex Roots

Equation: x² + x + 1 = 0

• D = 1² – 411 = -3
  
• Roots:
  

Root 1 = (-1 + i√3)/2  

Root 2 = (-1 – i√3)/2
Graph does not intersect x-axis; vertex = (-0.5, 0.75)

Frequently Asked Questions (FAQs)