(x-7)^x+1-(x-7)^x+11=0

3 min read Jun 17, 2024

Solving the Equation: (x-7)^x+1 - (x-7)^x+11 = 0

This equation presents a unique challenge due to the presence of variable exponents. Let's break down the steps to solve it:

1. Factor out a Common Term

Notice that both terms on the left-hand side share a common factor: (x-7)^x+1. We can factor this out:

(x-7)^x+1 [(1) - (x-7)^10] = 0

2. Apply the Zero Product Property

The Zero Product Property states that if the product of two factors is zero, then at least one of the factors must be zero. Therefore, we have two possible scenarios:

Scenario 1: (x-7)^x+1 = 0
Scenario 2: 1 - (x-7)^10 = 0

3. Solving Scenario 1:

Case 1.1: x - 7 = 0
- This leads to x = 7.
Case 1.2: x + 1 = 0
- This leads to x = -1.

Note: Remember that a number raised to a power cannot be equal to zero unless the base itself is zero.

4. Solving Scenario 2:

Isolate (x-7)^10:
- (x-7)^10 = 1
Take the tenth root of both sides:
- x-7 = 1
Solve for x:
- x = 8

5. Verification

It's crucial to verify if the potential solutions we found actually satisfy the original equation.

x = 7: The expression becomes 0 - 0 = 0. This solution works.
x = -1: The expression becomes ( -8 )^0 - ( -8 )^10 = 1 - 1073741824 = -1073741823. This solution doesn't work.
x = 8: The expression becomes (1)^9 - (1)^19 = 1 - 1 = 0. This solution works.

Conclusion

Therefore, the equation (x-7)^x+1 - (x-7)^x+11 = 0 has two solutions: x = 7 and x = 8.