1. What is the Quadratic Equation?

The quadratic equation solves for x in equations of the form ax² + bx + c = 0. It's a fundamental algebraic formula with applications across mathematics, physics, engineering, and economics.

2. How Does the Calculator Work?

The calculator uses the quadratic formula:

Where:

• \( a \) — Coefficient of x² (must not be zero)
• \( b \) — Coefficient of x
• \( c \) — Constant term
• \( \sqrt{b^2 - 4ac} \) — Discriminant (determines nature of roots)

Explanation: The discriminant determines the nature of the roots:

• Positive discriminant: Two distinct real roots
• Zero discriminant: One real root (repeated)
• Negative discriminant: Two complex conjugate roots

3. Importance of Quadratic Solutions

Details: Quadratic equations model many real-world phenomena including projectile motion, area problems, and optimization scenarios. Finding roots helps determine key points in these models.

4. Using the Calculator

Tips: Enter coefficients a, b, and c. All values must be numbers, with 'a' not equal to zero. The calculator handles real and complex roots.

5. Frequently Asked Questions (FAQ)

Q1: What if I get complex number results?
A: This means the equation has no real roots. The parabola doesn't intersect the x-axis. Complex roots come in conjugate pairs (a±bi).

Q2: What does a zero discriminant mean?
A: The parabola touches the x-axis at exactly one point (vertex). This represents a perfect square trinomial.

Q3: Can 'a' be zero?
A: No, if a=0 it becomes a linear equation (bx + c = 0) which has a different solution (x = -c/b).

Q4: How precise are the results?
A: Results are rounded to 4 decimal places. For exact solutions, use symbolic algebra systems.

Q5: What are common applications?
A: Used in physics (projectile motion), engineering (structural analysis), economics (profit maximization), and computer graphics.