1. What Is A Quadratic Equation?

A quadratic equation is a second-degree polynomial equation of the form ax² + bx + c = y, where a, b, and c are coefficients and a ≠ 0. It graphs as a parabola, a symmetrical U-shaped curve.

2. How Does The Calculator Work?

The calculator analyzes the quadratic equation:

Where:

• \( a \) — Coefficient of x² (determines parabola's direction and width)
• \( b \) — Coefficient of x (affects the position of the vertex)
• \( c \) — Constant term (y-intercept of the parabola)

Key Features Calculated: Vertex coordinates, discriminant, and real roots (if they exist).

3. Understanding Quadratic Graphs

Details: The graph of a quadratic equation is a parabola that opens upward if a > 0 and downward if a < 0. The vertex represents the minimum or maximum point, and the roots (if real) are the x-intercepts.

4. Using The Calculator

Tips: Enter the coefficients a, b, and c of your quadratic equation. The calculator will determine the vertex, discriminant, and roots. For accurate results, ensure a ≠ 0.

5. Frequently Asked Questions (FAQ)

Q1: What if the discriminant is negative?
A: A negative discriminant indicates the quadratic equation has no real roots (only complex roots).

Q2: What does the vertex represent?
A: The vertex is either the highest or lowest point on the parabola, representing the maximum or minimum value of the quadratic function.

Q3: Can a be zero in a quadratic equation?
A: No, if a = 0, the equation becomes linear (bx + c = y), not quadratic.

Q4: How does the value of a affect the graph?
A: If |a| > 1, the parabola is narrower; if 0 < |a| < 1, the parabola is wider. The sign of a determines direction (positive = upward, negative = downward).

Q5: What practical applications do quadratic equations have?
A: Quadratic equations model various real-world phenomena including projectile motion, area optimization, economics, and engineering problems.