Create a list of steps, in order, that will solve the following equation.
(x – 5)^2= 25
The Correct Answer and Explanation is :
To solve the equation ((x – 5)^2 = 25), follow these steps:
Steps to Solve the Equation
- Identify the Equation: Start with the given equation:
[
(x – 5)^2 = 25
] - Take the Square Root of Both Sides: To eliminate the square, take the square root of both sides. Remember to consider both the positive and negative square roots:
[
x – 5 = \pm 5
] - Separate into Two Equations: This will give you two separate equations to solve:
[
x – 5 = 5 \quad \text{(1)}
]
[
x – 5 = -5 \quad \text{(2)}
] - Solve the First Equation:
- For equation (1):
[
x – 5 = 5
]
Add 5 to both sides:
[
x = 10
]
- Solve the Second Equation:
- For equation (2):
[
x – 5 = -5
]
Add 5 to both sides:
[
x = 0
]
- List the Solutions: The solutions to the equation are:
[
x = 10 \quad \text{and} \quad x = 0
]
Explanation
The original equation ((x – 5)^2 = 25) involves a squared term, which indicates that we can use the property of square roots to simplify our work. Taking the square root of both sides introduces two possibilities because both a positive and a negative number, when squared, yield the same result (e.g., (5^2 = 25) and ((-5)^2 = 25)).
By separating the two cases after taking the square root, we ensure we capture all potential solutions for (x). The first case (x – 5 = 5) leads directly to (x = 10), while the second case (x – 5 = -5) results in (x = 0). Thus, both values represent solutions to the original equation, confirming that the process of squaring can introduce extraneous solutions, but in this case, both derived solutions are valid.
In conclusion, solving quadratic equations often involves reversing operations like squaring by applying square roots, and carefully managing the implications of those operations ensures that we find all potential solutions.