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(x-4)^2+(x-3)^2=25

2 min read Jun 17, 2024
(x-4)^2+(x-3)^2=25

Solving the Equation: (x-4)^2 + (x-3)^2 = 25

This equation represents a circle with a center at (4,3) and a radius of 5. Let's explore how to solve for the values of x that satisfy this equation.

Expanding and Simplifying

  1. Expand the squares: (x-4)^2 + (x-3)^2 = 25 x^2 - 8x + 16 + x^2 - 6x + 9 = 25

  2. Combine like terms: 2x^2 - 14x + 25 = 25

  3. Subtract 25 from both sides: 2x^2 - 14x = 0

  4. Factor out a 2x: 2x(x - 7) = 0

Finding the Solutions

  1. Set each factor to zero: 2x = 0 or x - 7 = 0

  2. Solve for x: x = 0 or x = 7

Conclusion

Therefore, the solutions to the equation (x-4)^2 + (x-3)^2 = 25 are x = 0 and x = 7. These values represent the x-coordinates of the points where the circle intersects the x-axis.

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