(x-5)^2=11

3 min read Jun 17, 2024

Solving the Equation (x-5)^2 = 11

This article will guide you through solving the quadratic equation (x-5)^2 = 11. We'll break down the steps and explore the solutions.

Understanding the Equation

The equation (x-5)^2 = 11 represents a quadratic equation. It's important to understand the following:

Square Root: The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 9 is 3 because 3 * 3 = 9.
Squaring: Squaring a number means multiplying it by itself. For example, 5 squared (5^2) is 5 * 5 = 25.

Solving the Equation

To solve for x, we need to isolate it. Here are the steps:

Take the square root of both sides: This will eliminate the square on the left side of the equation. Remember that taking the square root of a number can result in both positive and negative solutions. √((x-5)^2) = ±√11
Simplify: The square root of (x-5)^2 is simply (x-5). (x-5) = ±√11
Solve for x: Add 5 to both sides of the equation. x = 5 ±√11

Solutions

This gives us two solutions:

x = 5 + √11
x = 5 - √11

Approximating the Solutions

We can approximate the solutions using a calculator:

x ≈ 5 + 3.32 ≈ 8.32
x ≈ 5 - 3.32 ≈ 1.68

Conclusion

Therefore, the solutions to the equation (x-5)^2 = 11 are x = 5 + √11 and x = 5 - √11, which are approximately 8.32 and 1.68, respectively.