1. What is the Quadratic Formula?

The quadratic formula provides the solution(s) to a quadratic equation of the form ax² + bx + c = 0. It's derived from completing the square and works for all quadratic equations, including those with complex roots.

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

Explanation: The discriminant (b² - 4ac) determines the nature of the roots:

• Positive discriminant: Two distinct real roots
• Zero discriminant: One real double root
• Negative discriminant: Two complex conjugate roots

3. Understanding the Results

Details: The calculator provides:

• The roots (solutions) of the equation
• The factored form of the quadratic (when possible with real coefficients)
• The discriminant value and its interpretation

4. Using the Calculator

Tips: Enter the coefficients a, b, and c from your quadratic equation. The coefficient a must not be zero (otherwise it's not quadratic).

5. Frequently Asked Questions (FAQ)

Q1: What if I get complex roots?
A: Complex roots occur when the discriminant is negative. They come in conjugate pairs (a±bi) and are valid mathematical solutions.

Q2: Why can't some quadratics be factored?
A: Only quadratics with real roots can be factored into real binomials. Complex roots require complex numbers for factoring.

Q3: What does a zero discriminant mean?
A: It means the quadratic has exactly one real solution (a "double root") and its graph touches the x-axis at exactly one point.

Q4: Can I use this for any quadratic equation?
A: Yes, as long as a ≠ 0. The formula works for all cases - real roots, double roots, and complex roots.

Q5: How accurate are the results?
A: Results are calculated to high precision (4 decimal places shown) but may be limited by floating-point arithmetic in extreme cases.