(x+1)(x+2)(x+3)(x+4)=120 Solve The Equation

Jasonbradley

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Jun 16, 2024

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

This equation involves a product of four consecutive terms equaling a constant. To solve it, we can employ a combination of algebraic manipulation and strategic thinking.

1. Recognizing the Pattern:

The left-hand side of the equation represents the product of four consecutive integers. We can use this pattern to our advantage.

2. Factoring 120:

Start by factoring 120 into its prime factors: 120 = 2 x 2 x 2 x 3 x 5

3. Finding Consecutive Integers:

Now, try to group these prime factors in a way that forms four consecutive integers. Observe that:

• 2 x 3 = 6
• 2 x 5 = 10

These are two consecutive integers. The other two consecutive integers can be formed by subtracting 1 and adding 1:

• 6 - 1 = 5
• 10 + 1 = 11

Therefore, we have found the four consecutive integers: 5, 6, 10, and 11.

4. Verification:

Let's verify our solution:

(5 + 1)(5 + 2)(5 + 3)(5 + 4) = 6 x 7 x 8 x 9 = 3024

This doesn't match the original equation. This indicates that our initial assumption that the product of four consecutive integers must be equal to 120 is incorrect.

5. Finding the Solution:

Since the initial assumption was incorrect, we need to solve the equation directly.

Here's how we can solve the equation (x+1)(x+2)(x+3)(x+4) = 120:

• Expand the product: Expand the left side of the equation. This will result in a fourth-degree polynomial.
• Rearrange: Move all terms to one side to get a polynomial equation equal to zero.
• Factoring: Attempt to factor the polynomial. This might be challenging, but using techniques like the Rational Root Theorem can be helpful.
• Solve for x: Once you have factored the polynomial, set each factor equal to zero and solve for the individual values of x.

Note: The equation (x+1)(x+2)(x+3)(x+4) = 120 might not have rational solutions. You might need to use numerical methods or graphing techniques to find the approximate solutions.

In conclusion:

While the initial approach of finding consecutive integers might seem promising, it does not yield the correct solution. To solve the equation (x+1)(x+2)(x+3)(x+4) = 120, you will need to expand the equation, rearrange it, and attempt to factor the resulting polynomial.