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(x^2+2x-5)^2=(x^2-x+5)^2

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

Solving the Equation (x^2 + 2x - 5)^2 = (x^2 - x + 5)^2

This equation presents a unique challenge due to the presence of squares on both sides. Let's break down the solution step by step:

1. Simplify using the Square Root Property

Since both sides are squared, we can take the square root of both sides, resulting in two possible equations:

  • Equation 1: x^2 + 2x - 5 = x^2 - x + 5
  • Equation 2: x^2 + 2x - 5 = -(x^2 - x + 5)

2. Solve Equation 1

  • Combine like terms: 3x = 10
  • Solve for x: x = 10/3

3. Solve Equation 2

  • Distribute the negative sign: x^2 + 2x - 5 = -x^2 + x - 5
  • Combine like terms: 2x^2 + x = 0
  • Factor out x: x(2x + 1) = 0
  • Solve for x:
    • x = 0
    • 2x + 1 = 0 --> x = -1/2

4. Verify Solutions

It's crucial to verify our solutions by plugging them back into the original equation:

  • x = 10/3 satisfies the original equation.
  • x = 0 satisfies the original equation.
  • x = -1/2 satisfies the original equation.

Conclusion

Therefore, the solutions to the equation (x^2 + 2x - 5)^2 = (x^2 - x + 5)^2 are x = 10/3, x = 0, and x = -1/2.

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