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Solving the Equation: (x + 5)² = 11
This equation presents a straightforward example of solving a quadratic equation. Here's how we can approach it:
1. Understanding the Equation
The equation (x + 5)² = 11 is a quadratic equation because it involves a variable raised to the power of 2.
2. Isolate the Squared Term
To solve for x, we need to isolate the term (x + 5)². We can do this by taking the square root of both sides of the equation.
- √[(x + 5)²] = ±√11
Notice we include both positive and negative square roots because squaring a positive or negative number results in a positive value.
3. Simplify and Solve for x
- x + 5 = ±√11
- x = -5 ±√11
4. The Solutions
Therefore, the solutions to the equation (x + 5)² = 11 are:
- x = -5 + √11
- x = -5 - √11
These are the two distinct real solutions to the equation.
5. Verification
We can always verify our solutions by plugging them back into the original equation. Let's try one of them:
-
x = -5 + √11
-
(-5 + √11 + 5)² = (√11)² = 11
As you can see, the solution satisfies the original equation.
Conclusion
Solving quadratic equations like (x + 5)² = 11 involves understanding the properties of squares, applying the square root operation, and carefully considering both positive and negative possibilities.