1. What is Quadratic Factoring?

Quadratic factoring is the process of breaking down a quadratic equation into simpler multiplicative components (factors). The standard form of a quadratic equation is ax² + bx + c = 0, which can be factored into the form a(x - r₁)(x - r₂) where r₁ and r₂ are the 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. When positive, there are two real roots; when zero, one real root; when negative, complex conjugate roots.

3. Importance of Quadratic Equations

Details: Quadratic equations appear throughout mathematics and physics, describing parabolic motion, optimization problems, and many natural phenomena. Factoring helps find roots which represent key points in these applications.

4. Using the Calculator

Tips: Enter coefficients a, b, and c. The calculator will find the roots and display the factored form. Remember a cannot be zero (linear equation).

5. Frequently Asked Questions (FAQ)

Q1: What if I get complex roots?
A: Complex roots indicate the parabola doesn't intersect the x-axis. The factored form still holds mathematically, though it can't be graphed on real axes.

Q2: Why does a=1 give simpler factored form?
A: When a=1, the equation is monic and the factored form doesn't need the leading coefficient.

Q3: What if the discriminant is zero?
A: A zero discriminant means one real root (a perfect square), and the factored form shows the squared term (x - r)².

Q4: Can this solve all quadratics?
A: Yes, the quadratic formula provides solutions to all quadratic equations, including those that can't be factored by inspection.

Q5: How is this different from completing the square?
A: Both methods find roots, but factoring by grouping is often easier for simple equations while the quadratic formula works universally.