🔢 Factoring Calculator
Factor quadratics, find prime factors, and calculate GCF/LCM with step-by-step solutions.
Factor Quadratic: ax² + bx + c
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Factored Form
Prime Factorization
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Prime Factorization
Greatest Common Factor & LCM
Factoring Tips
- Quadratic formula: x = (−b ± √(b²−4ac)) / 2a
- If discriminant b²−4ac < 0: no real factors
- GCF(a,b) × LCM(a,b) = a × b
- Use Euclidean algorithm for efficient GCF calculation
How the Factoring Calculator Works
How Factoring Works
Factoring breaks down a polynomial into simpler expressions (factors) that multiply together to give the original polynomial. This process is essential for solving equations, simplifying expressions, and finding roots.
Common Factoring Patterns:
Greatest Common Factor (GCF):
6x² + 9x = 3x(2x + 3)
Always check for GCF first
6x² + 9x = 3x(2x + 3)
Always check for GCF first
Difference of Squares:
x² – 16 = (x + 4)(x – 4)
Pattern: a² – b² = (a+b)(a-b)
x² – 16 = (x + 4)(x – 4)
Pattern: a² – b² = (a+b)(a-b)
Perfect Square Trinomial:
x² + 6x + 9 = (x + 3)²
Pattern: a² ± 2ab + b² = (a ± b)²
x² + 6x + 9 = (x + 3)²
Pattern: a² ± 2ab + b² = (a ± b)²
General Trinomial:
x² + 5x + 6 = (x + 2)(x + 3)
Find two numbers that multiply to c and add to b
x² + 5x + 6 = (x + 2)(x + 3)
Find two numbers that multiply to c and add to b
Real-World Applications:
- Engineering: Structural analysis, load distribution calculations
- Physics: Motion equations, energy calculations, wave functions
- Economics: Cost-revenue optimization, break-even analysis
- Computer Science: Algorithm optimization, cryptography
- Architecture: Area calculations, design optimization
Step-by-Step Strategy:
- Look for and factor out the Greatest Common Factor (GCF)
- Count the terms: 2 terms → difference of squares, 3 terms → trinomial
- Check for special patterns (perfect squares, difference of squares)
- For trinomials: find factors of ‘ac’ that add to ‘b’
- Verify by expanding the factored form