1. What is Standard Factored Form?

The standard factored form of a quadratic equation is expressed as \( f(x) = a(x - r_1)(x - r_2) \), where \( a \) is the leading coefficient, and \( r_1 \) and \( r_2 \) are the roots of the equation. This form clearly shows the x-intercepts of the parabola.

2. How Does the Calculator Work?

The calculator uses the standard factored form:

Where:

• \( a \) — Leading coefficient (determines the parabola's width and direction)
• \( r_1 \) — First root (x-intercept)
• \( r_2 \) — Second root (x-intercept)

Explanation: The calculator converts the factored form to expanded form by multiplying out the factors and combining like terms.

3. Importance of Factored Form

Details: Factored form is particularly useful for quickly identifying the roots of the equation and understanding the graph's x-intercepts. It's also helpful for solving quadratic equations.

4. Using the Calculator

Tips: Enter the leading coefficient (a) and both roots (r₁ and r₂). The calculator will display both the factored form and the expanded standard form of the quadratic equation.

5. Frequently Asked Questions (FAQ)

Q1: What if my quadratic has complex roots?
A: This calculator works with real roots. For complex roots, the factored form would involve complex numbers.

Q2: What if there's only one real root?
A: For a double root, enter the same value for both r₁ and r₂ (the factored form would show \( (x - r)^2 \)).

Q3: Can I use fractions as input?
A: Yes, you can enter decimal equivalents of fractions (e.g., 0.5 for 1/2).

Q4: How does the leading coefficient affect the graph?
A: A positive 'a' opens upward, negative opens downward. Larger absolute values make the parabola narrower.

Q5: What's the difference between factored form and vertex form?
A: Factored form shows roots, while vertex form \( f(x) = a(x - h)^2 + k \) shows the vertex (h,k) directly.