(x-4)2=169 Extracting Square Roots

Jasonbradley

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Jun 17, 2024

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Solving for x: Extracting Square Roots

In this article, we will walk through the steps of solving the equation (x-4)² = 169 using the method of extracting square roots.

Understanding the Equation

The equation (x-4)² = 169 is a quadratic equation, meaning it involves a variable raised to the power of 2. To solve for 'x', we need to isolate it.

Extracting Square Roots

1. Isolate the squared term: The term (x-4)² is already isolated on the left side of the equation.

2. Take the square root of both sides: Taking the square root of both sides of the equation cancels out the square on the left side. Remember, when taking the square root, we need to consider both positive and negative solutions.

   √(x-4)² = ±√169

3. Simplify: The square root of (x-4)² is simply (x-4). The square root of 169 is 13.

   x - 4 = ±13

4. Solve for x: To isolate x, add 4 to both sides of the equation:

   x = 4 ± 13

5. Final Solutions: This gives us two possible solutions:

   • x = 4 + 13 = 17
   • x = 4 - 13 = -9

Conclusion

By extracting square roots, we have successfully solved the equation (x-4)² = 169, obtaining two solutions: x = 17 and x = -9. This method offers a straightforward and efficient approach to solving quadratic equations where the squared term is already isolated.