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(x-2)2+(x+1)2=5

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

Solving the Equation (x-2)² + (x+1)² = 5

This equation represents a quadratic equation in disguise. Let's break it down and solve for the values of 'x'.

Expanding and Simplifying

  1. Expand the squares:

    • (x-2)² = x² - 4x + 4
    • (x+1)² = x² + 2x + 1

  2. Substitute back into the original equation:

    • (x² - 4x + 4) + (x² + 2x + 1) = 5

  3. Combine like terms:

    • 2x² - 2x + 5 = 5

  4. Simplify further:

    • 2x² - 2x = 0

Solving the Quadratic Equation

We now have a simplified quadratic equation. There are a few ways to solve this:

  • Factoring:

    • Factor out a 2x: 2x(x - 1) = 0
    • Set each factor equal to zero: 2x = 0 or x - 1 = 0
    • Solve for x: x = 0 or x = 1

  • Quadratic Formula:

    • The quadratic formula is a general solution for any quadratic equation in the form ax² + bx + c = 0:
    • x = (-b ± √(b² - 4ac)) / 2a
    • In our case, a = 2, b = -2, and c = 0.
    • Substitute these values into the formula and you'll arrive at the same solutions as factoring: x = 0 or x = 1.

Conclusion

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

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