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(x-2)^2-(x-3)^2=7-2(5-x)

2 min read Jun 17, 2024
(x-2)^2-(x-3)^2=7-2(5-x)

Solving the Equation (x-2)^2 - (x-3)^2 = 7 - 2(5-x)

This article will guide you through the steps of solving the equation (x-2)^2 - (x-3)^2 = 7 - 2(5-x).

1. Expanding the Squares

First, we need to expand the squares on both sides of the equation using the formula (a-b)^2 = a^2 - 2ab + b^2.

  • Left-hand side:

    • (x-2)^2 = x^2 - 4x + 4
    • (x-3)^2 = x^2 - 6x + 9
    • Therefore, (x-2)^2 - (x-3)^2 = (x^2 - 4x + 4) - (x^2 - 6x + 9) = 2x - 5

  • Right-hand side:

    • 7 - 2(5-x) = 7 - 10 + 2x = 2x - 3

2. Simplifying the Equation

Now, our equation becomes 2x - 5 = 2x - 3.

3. Solving for x

Notice that both sides of the equation have the same term 2x. Subtracting 2x from both sides gives:

  • -5 = -3

This is a contradiction. Therefore, the equation (x-2)^2 - (x-3)^2 = 7 - 2(5-x) has no solution.

Conclusion

The equation (x-2)^2 - (x-3)^2 = 7 - 2(5-x) has no solutions. This is because simplifying the equation leads to a contradiction.

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