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(x-7)^2+(x-2)^2=10

2 min read Jun 17, 2024
(x-7)^2+(x-2)^2=10

Solving the Equation (x-7)² + (x-2)² = 10

This article will guide you through solving the equation (x-7)² + (x-2)² = 10. This equation represents a quadratic equation in disguise, and we'll use algebraic manipulation and the quadratic formula to find the solutions.

Expanding and Simplifying

  1. Expand the squares: (x-7)² = (x-7)(x-7) = x² - 14x + 49 (x-2)² = (x-2)(x-2) = x² - 4x + 4

  2. Substitute the expanded terms back into the equation: x² - 14x + 49 + x² - 4x + 4 = 10

  3. Combine like terms: 2x² - 18x + 53 = 10

  4. Move all terms to one side to form a standard quadratic equation: 2x² - 18x + 43 = 0

Solving the Quadratic Equation

Now we have a standard quadratic equation in the form ax² + bx + c = 0, where a = 2, b = -18, and c = 43.

We can use the quadratic formula to solve for x:

x = (-b ± √(b² - 4ac)) / 2a

  1. Substitute the values of a, b, and c: x = (18 ± √((-18)² - 4 * 2 * 43)) / (2 * 2)

  2. Simplify: x = (18 ± √(324 - 344)) / 4 x = (18 ± √(-20)) / 4 x = (18 ± 2√5i) / 4

  3. Reduce the fraction: x = 9/2 ± √5i / 2

Therefore, the solutions to the equation (x-7)² + (x-2)² = 10 are x = 9/2 + √5i / 2 and x = 9/2 - √5i / 2. These solutions are complex numbers, indicating that the equation does not have real roots.

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