(x+1)(x+2)(x+3)(x+4)=120

3 min read Jun 16, 2024

Solving the Equation (x+1)(x+2)(x+3)(x+4) = 120

This equation presents a unique challenge as it involves a product of four binomials. Here's how we can approach solving it:

1. Understanding the Equation

The equation (x+1)(x+2)(x+3)(x+4) = 120 is a quartic equation, meaning it has a highest power of x as 4. Finding its solutions can be a bit more complex than solving linear or quadratic equations.

2. Factoring the Right-Hand Side

Start by factoring the right-hand side of the equation: 120 = 2⁴ * 3 * 5

3. Observing the Pattern

Notice that the left-hand side of the equation consists of consecutive integers. This suggests that we might be able to find a set of consecutive integers that multiply to 120.

4. Trial and Error

Let's try a few combinations of consecutive integers:

2 * 3 * 4 * 5 = 120

This combination satisfies the equation, therefore:

x + 1 = 2
x + 2 = 3
x + 3 = 4
x + 4 = 5

Solving for x in any of these equations gives us: x = 1

5. Important Note

While we found one solution, there might be other solutions. However, since we're dealing with a quartic equation, there could be up to four solutions. To find all potential solutions, we would need to expand the left-hand side of the equation, move the constant term to the left, and then employ techniques for solving quartic equations. This process can be quite complex and might require advanced mathematical tools.

Conclusion

We successfully found one solution to the equation (x+1)(x+2)(x+3)(x+4) = 120, which is x = 1. While there might be other solutions, finding them requires more advanced mathematical techniques. This problem highlights the importance of recognizing patterns and applying creative problem-solving strategies in mathematics.