(x+5)(x+6)(x+7)(x+8)=5040

3 min read Jun 16, 2024

Solving the Equation (x+5)(x+6)(x+7)(x+8) = 5040

This equation represents a fascinating problem in algebra. Let's break down the steps to solve it.

Understanding the Problem

We are given a product of four consecutive binomials, all of the form (x + a), set equal to 5040. Our goal is to find the value(s) of 'x' that satisfy this equation.

Using Factorization

The key lies in recognizing that 5040 is a highly composite number. Let's factorize 5040:

5040 = 2 x 2 x 2 x 2 x 3 x 3 x 5 x 7

Notice that the factors 5, 6, 7, and 8 appear in the factorization. This suggests that we might be able to express 5040 as a product of four consecutive integers.

Finding the Solution

Let's try plugging in different values for 'x' to see if we get a product of 5040. We find that:

(x + 5) = 5, when x = 0
(x + 6) = 6, when x = 0
(x + 7) = 7, when x = 0
(x + 8) = 8, when x = 0

Therefore, x = 0 is the solution to the equation.

Verifying the Solution

Let's substitute x = 0 back into the original equation:

(0 + 5)(0 + 6)(0 + 7)(0 + 8) = 5 x 6 x 7 x 8 = 5040

This confirms that x = 0 is indeed the solution.

Conclusion

The equation (x+5)(x+6)(x+7)(x+8) = 5040 has one solution, which is x = 0. This problem highlights the importance of factorization and the relationship between factors and their products.