Question
22. Higher Order Thinking a. Solve \( (x-5)^{2}-100=0 \). Show the steps for yoution.
Ask by Lindsey Clarke. in the United States
Mar 10,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The solutions are \( x = 15 \) and \( x = -5 \).
Solution
1. Start with the equation:
\[
(x-5)^2 - 100 = 0
\]
2. Add \(100\) to both sides:
\[
(x-5)^2 = 100
\]
3. Take the square root of both sides (remembering to consider both positive and negative roots):
\[
x-5 = \pm \sqrt{100}
\]
\[
x-5 = \pm 10
\]
4. Solve for \(x\) for each case:
- If \(x-5 = 10\):
\[
x = 10 + 5 = 15
\]
- If \(x-5 = -10\):
\[
x = -10 + 5 = -5
\]
5. The solutions are:
\[
x = 15 \quad \text{or} \quad x = -5
\]
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Simplify this solution The Deep Dive
To solve the equation \( (x-5)^{2}-100=0 \), start by isolating the squared term. Add 100 to both sides to get \( (x-5)^{2}=100 \). Then, take the square root of both sides: \( x-5 = \pm 10 \). This gives us two equations: \( x-5=10 \) and \( x-5=-10 \). Solving these, we find \( x=15 \) and \( x=-5 \). So the solutions are \( x=15 \) and \( x=-5 \). Understanding the process behind higher-order thinking in mathematics allows for deeper problem-solving strategies. For instance, relating similar problems and recognizing patterns in equations can enhance your ability to tackle more complex challenges in algebra and beyond!
