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(x-4)^2-49=0

2 min read Jun 17, 2024
(x-4)^2-49=0

Solving the Equation (x-4)^2 - 49 = 0

This article will guide you through the steps of solving the quadratic equation (x-4)^2 - 49 = 0.

Understanding the Equation

The equation (x-4)^2 - 49 = 0 represents a quadratic equation in standard form. Here's why:

  • Quadratic: The highest power of the variable 'x' is 2 (from the term (x-4)^2).
  • Standard Form: The equation is arranged in the form ax^2 + bx + c = 0, where a, b, and c are constants.

Solving for x

To solve for x, we can use the following steps:

  1. Isolate the squared term:

    • Add 49 to both sides of the equation: (x-4)^2 = 49

  2. Take the square root of both sides:

    • Remember to include both positive and negative roots: x - 4 = ±√49

  3. Simplify:

    • √49 = 7 x - 4 = ±7

  4. Solve for x:

    • Add 4 to both sides: x = 4 ± 7

  5. Find the two solutions:

    • x = 4 + 7 = 11
    • x = 4 - 7 = -3

Solutions

Therefore, the solutions to the equation (x-4)^2 - 49 = 0 are x = 11 and x = -3.

Verification

We can verify our solutions by substituting them back into the original equation:

  • For x = 11: (11 - 4)^2 - 49 = 7^2 - 49 = 49 - 49 = 0
  • For x = -3: (-3 - 4)^2 - 49 = (-7)^2 - 49 = 49 - 49 = 0

Both solutions satisfy the equation, confirming our results.

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