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(x-5)^2-16=0

2 min read Jun 17, 2024
(x-5)^2-16=0

Solving the Equation (x-5)^2 - 16 = 0

This article will guide you through solving the equation (x-5)^2 - 16 = 0. This equation is a quadratic equation in disguise, and we can solve it using a couple of methods.

Method 1: Using the Square Root Property

  1. Isolate the squared term: Add 16 to both sides of the equation: (x-5)^2 = 16

  2. Take the square root of both sides: Remember that taking the square root introduces both positive and negative solutions: x - 5 = ±4

  3. Solve for x:

    • x - 5 = 4 => x = 9
    • x - 5 = -4 => x = 1

Therefore, the solutions to the equation (x-5)^2 - 16 = 0 are x = 9 and x = 1.

Method 2: Expanding and Solving the Quadratic

  1. Expand the square: (x - 5)^2 = x^2 - 10x + 25

  2. Rewrite the equation: x^2 - 10x + 25 - 16 = 0 x^2 - 10x + 9 = 0

  3. Factor the quadratic: (x - 9)(x - 1) = 0

  4. Solve for x:

    • x - 9 = 0 => x = 9
    • x - 1 = 0 => x = 1

Again, we arrive at the solutions x = 9 and x = 1.

Conclusion

Both methods demonstrate that the solutions to the equation (x-5)^2 - 16 = 0 are x = 9 and x = 1. You can choose the method that you find more comfortable or efficient.

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