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(x-1 5)^6+2(1 5-x)^2=0

2 min read Jun 17, 2024
(x-1 5)^6+2(1 5-x)^2=0

Solving the Equation: (x - 15)^6 + 2(15 - x)^2 = 0

This equation presents a unique challenge due to the presence of both a sixth power term and a squared term. We can use a few key algebraic manipulations to solve for the value of x.

Simplifying the Equation

  1. Recognize the pattern: Notice that the terms (x - 15) and (15 - x) are essentially the same, just with opposite signs. We can rewrite the second term using the property (-1)^2 = 1:

    (x - 15)^6 + 2(-1)^2 (x - 15)^2 = 0

  2. Factor out the common term: Both terms share (x - 15)^2 as a common factor. Let's factor it out:

    (x - 15)^2 [(x - 15)^4 + 2] = 0

Solving for x

Now we have a product of two factors that equals zero. This means at least one of the factors must be equal to zero:

  1. Factor 1: (x - 15)^2 = 0

    • Taking the square root of both sides: x - 15 = 0
    • Solving for x: x = 15

  2. Factor 2: (x - 15)^4 + 2 = 0

    • Subtracting 2 from both sides: (x - 15)^4 = -2
    • This equation has no real solutions, as any real number raised to an even power cannot be negative.

Conclusion

Therefore, the only real solution to the equation (x - 15)^6 + 2(15 - x)^2 = 0 is x = 15.

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