Generic Rectangle Calculator
Factor quadratic equations using the generic rectangle method with step-by-step solutions
Factor Quadratic Equation: ax² + bx + c = 0
Cannot be zero for quadratic equation
Linear term coefficient
Constant term
Factorization Results
Generic Rectangle
Green cells: x² and constant terms (placed diagonally)
Yellow cells: Middle term factors
Example Calculation
Factor: 4x² + 12x + 5 = 0
Step 1: Place 4x² (bottom-left) and 5 (top-right) diagonally
Step 2: Calculate product: 4x² × 5 = 20x²
Step 3: Find factors of 20: 1×20, 2×10, 4×5
Step 4: Choose pair that adds to 12: 10 + 2 = 12
Step 5: Place 10x and 2x in remaining cells
Step 6: Find common factors: (2x + 5)(2x + 1)
Result
Factorization: (2x + 5)(2x + 1) = 0
Solutions: x = -5/2, x = -1/2
Generic Rectangle Method
Place the x² term and constant term at opposite diagonal corners
Find the product of these diagonal terms
List all factor pairs of the product coefficient
Choose the pair that adds up to the middle term coefficient
Find common factors to get the final factorization
Common Examples
x² + 3x + 2
= (x + 1)(x + 2)
x² - 5x + 6
= (x - 2)(x - 3)
2x² + 7x + 3
= (2x + 1)(x + 3)
x² - 4
= (x - 2)(x + 2)
Understanding Generic Rectangles
What are Generic Rectangles?
Generic rectangles are visual tools used in mathematics to factor quadratic expressions. The word "generic" refers to the fact that these rectangles are not drawn to scale - they're conceptual diagrams that help organize the factoring process.
Why Use This Method?
- •Visual approach makes factoring more intuitive
- •Systematic method reduces errors
- •Works for all factorizable quadratics
- •Helps understand the structure of quadratic expressions
The Process
Step 1: Setup
Create a 2×2 grid and place the x² term and constant term diagonally
Step 2: Product
Calculate the product of the diagonal terms
Step 3: Factors
Find factor pairs that add to the middle coefficient
Step 4: Complete
Find common factors to get the final answer