1. What is the Quadratic Formula?

The quadratic formula is a fundamental algebraic formula that provides the solutions to quadratic equations of the form ax² + bx + c = 0. It is one of the most important formulas in algebra and has wide applications in mathematics, physics, engineering, and economics.

2. How Does the Calculator Work?

The calculator uses the quadratic formula:

Where:

• \( a \) — Coefficient of x² (cannot be zero)
• \( b \) — Coefficient of x
• \( c \) — Constant term
• \( \pm \) — Indicates two possible solutions

Explanation: The expression under the square root (b² - 4ac) is called the discriminant, which 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. Applications of Quadratic Equations

Details: Quadratic equations appear in various real-world scenarios including projectile motion, optimization problems, area calculations, economics (profit maximization), and engineering design problems.

4. Using the Calculator

Tips: Enter the coefficients a, b, and c from your quadratic equation ax² + bx + c = 0. The calculator will provide real or complex solutions as appropriate. Remember that coefficient 'a' cannot be zero.

5. Frequently Asked Questions (FAQ)

Q1: What if I get complex solutions?
A: Complex solutions (containing 'i') occur when the discriminant is negative, meaning the parabola doesn't cross the x-axis. These are valid mathematical solutions.

Q2: Why can't coefficient 'a' be zero?
A: If a = 0, the equation becomes linear (bx + c = 0), not quadratic. The quadratic formula is specifically for second-degree polynomials.

Q3: How accurate are the solutions?
A: Solutions are calculated with high precision (4 decimal places), but exact solutions may involve irrational numbers that require rounding.

Q4: Can I solve equations with fractions or decimals?
A: Yes, the calculator accepts decimal coefficients. For fractional coefficients, you can convert them to decimals or find a common denominator.

Q5: What's the geometric interpretation?
A: The solutions represent the x-intercepts (roots) of the parabola y = ax² + bx + c, where the graph crosses the x-axis.