Particular Solution Calculator

Using the Particular Solution Calculator

This guide will walk you through using the Particular Solution Calculator to determine the discriminant, solutions, and vertex of a quadratic equation in the form ax² + bx + c. Follow the steps below to make full use of the calculator’s features.

Step 1: Enter the Coefficient Values

• Coefficient a: Enter the value of coefficient a in the equation ax² + bx + c.
  • This field is required.
  • You may enter any real number.
• Coefficient b: Enter the value of coefficient b in the equation ax² + bx + c.
  • This field is required.
  • You may enter any real number.
• Coefficient c: Enter the value of coefficient c in the equation ax² + bx + c.
  • This field is required.
  • You may enter any real number.

Step 2: Calculate and Interpret the Results

• Discriminant (Δ): The calculator will compute the discriminant using the formula Δ = b² – 4ac. This value will help determine the nature of the roots.
  • If Δ > 0, the equation has two distinct real roots.
  • If Δ = 0, the equation has exactly one real root (a repeated root).
  • If Δ < 0, the equation has no real roots and two complex roots.
• First Solution (x₁): The calculator will find the first solution using the formula x₁ = (-b + √Δ) / (2a).
• Second Solution (x₂): The calculator will find the second solution using the formula x₂ = (-b – √Δ) / (2a).
• Vertex X-coordinate: The calculator will compute the x-coordinate of the vertex using the formula x = -b / (2a). This represents the axis of symmetry of the parabola.
• Vertex Y-coordinate: The calculator will compute the y-coordinate of the vertex using the formula y = -Δ / (4a). This represents the maximum or minimum value of the quadratic function.

Carefully review the results, particularly the discriminant, to understand the nature of the solutions. The solutions x₁ and x₂ provide the roots of the equation, while the vertex coordinates give insight into the parabola’s shape and position relative to the coordinate plane.