Quadratic Fraction Calculator

Quadratic Fraction Decomposition:

\[ \frac{P(x)}{(ax^2 + bx + c)(dx^2 + ex + f)} = \frac{Ax + B}{ax^2 + bx + c} + \frac{Cx + D}{dx^2 + ex + f} \]

Partial Fraction Decomposition:

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1. What Is Quadratic Fraction Decomposition?

Quadratic fraction decomposition is a mathematical technique used to break down complex rational expressions into simpler partial fractions when the denominator contains irreducible quadratic factors. This method is particularly useful in calculus for integration and in solving differential equations.

2. How Does The Calculator Work?

The calculator uses algebraic decomposition algorithm:

\[ \frac{P(x)}{Q(x)R(x)} = \frac{Ax + B}{Q(x)} + \frac{Cx + D}{R(x)} \]

Where:

\( P(x) \) — Numerator polynomial
\( Q(x), R(x) \) — Quadratic factors in denominator
\( A, B, C, D \) — Constants to be determined

Explanation: The algorithm solves a system of linear equations to find the constants that make the decomposition valid for all x.

3. Importance Of Partial Fractions

Details: Partial fraction decomposition is essential for integrating rational functions, solving linear differential equations with constant coefficients, and simplifying complex algebraic expressions in engineering and physics applications.

4. Using The Calculator

Tips: Enter the numerator polynomial and both quadratic denominator factors using standard polynomial notation (e.g., x^2 + 3x - 4). Ensure the degree of numerator is less than the degree of denominator for proper decomposition.

5. Frequently Asked Questions (FAQ)

Q1: What if the denominator has repeated quadratic factors?
A: For repeated factors, additional terms with higher powers in the denominator are needed in the decomposition.

Q2: Can this handle denominators with linear factors?
A: Yes, the calculator can handle mixed factors, but the current implementation focuses on quadratic factors.

Q3: What polynomial formats are supported?
A: Standard algebraic notation with coefficients, variables, and exponents using ^ symbol.

Q4: Are there limitations to this decomposition?
A: The method requires that the denominator factors are irreducible over the reals and that the numerator degree is appropriate.

Q5: How is the system of equations solved?
A: The calculator uses matrix methods to solve the linear system that results from equating coefficients of like powers.