(x%2B3)(x%2B5)(x%2B7)%3D5760.jpeg "(x+1)(x+3)(x+5)(x+7)=5760")
Solving the Equation (x+1)(x+3)(x+5)(x+7) = 5760
This problem involves solving a quartic equation, which can be quite challenging. However, we can use some clever techniques to simplify it and find the solution(s).
Recognizing the Pattern and Simplification
Notice the pattern in the factors: (x+1), (x+3), (x+5), (x+7). They are all consecutive odd numbers. This pattern suggests we can try to manipulate the equation to make it easier to solve.
Step 1: Manipulate the factors
Let's introduce a new variable, y = x + 3. This allows us to rewrite the equation:
(x + 1)(x + 3)(x + 5)(x + 7) = 5760
(y - 2)(y)(y + 2)(y + 4) = 5760
Step 2: Simplify the expression
Now, we can group the terms:
(y^2 - 4)(y^2 + 4y) = 5760
Step 3: Expand and Rearrange
Expand the expression and move all terms to one side:
y^4 + 4y^3 - 4y^2 - 16y - 5760 = 0
Solving the Quartic Equation
We now have a quartic equation in terms of y. Unfortunately, there's no general formula to solve quartic equations directly. However, we can try to factor it or use numerical methods to find the solutions.
Step 4: Factorization (Optional)
In this case, we can factor the equation:
(y - 12)(y + 12)(y^2 + 4y + 40) = 0
This gives us three possible solutions:
- y = 12
- y = -12
- y^2 + 4y + 40 = 0
The quadratic equation y^2 + 4y + 40 = 0 has no real solutions.
Step 5: Finding the values of x
Since we defined y = x + 3, we can find the values of x:
- y = 12: x = 12 - 3 = 9
- y = -12: x = -12 - 3 = -15
Therefore, the solutions to the equation (x+1)(x+3)(x+5)(x+7)=5760 are:
x = 9 and x = -15.
Important Note
The factorization in step 4 might not always be easy to find. For more complex quartic equations, you might need to use numerical methods or software tools to approximate the solutions.