(x+1)(x+2)(x+3)(x+4)=840

2 min read Jun 16, 2024

Solving the Equation (x+1)(x+2)(x+3)(x+4) = 840

This equation presents an interesting challenge due to its form. Let's explore how to solve it:

Understanding the Equation

The equation represents the product of four consecutive integers equaling 840. This hints at a potential solution involving factorization.

Factorization Approach

Prime Factorization of 840: Begin by finding the prime factorization of 840: 840 = 2 x 2 x 2 x 3 x 5 x 7.
Grouping for Consecutive Integers: Look for combinations of these prime factors that can be grouped to form four consecutive integers. Notice that 5 x 6 x 7 x 8 = 840.
Solution: Therefore, we have:
- x + 1 = 5
- x + 2 = 6
- x + 3 = 7
- x + 4 = 8
- Solving for x, we get x = 4.

Verification

Substituting x = 4 back into the original equation: (4 + 1)(4 + 2)(4 + 3)(4 + 4) = 5 x 6 x 7 x 8 = 840. This confirms our solution.

Conclusion

By utilizing prime factorization and careful observation, we successfully found the solution to the equation (x + 1)(x + 2)(x + 3)(x + 4) = 840. This approach demonstrates the importance of recognizing patterns and applying appropriate techniques to solve algebraic equations.