Question
Use the square root property to solve the equation. \[ (x-5)^{2}=3 \]
Ask by Christensen Lindsey. in the United States
Mar 01,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The solutions are \( x = 5 + \sqrt{3} \) and \( x = 5 - \sqrt{3} \).
Solution
To solve the equation (x – 5)² = 3 using the square root property, follow these steps:
1. Take the square root of both sides:
√[(x – 5)²] = ±√3
2. This simplifies to:
x – 5 = ±√3
3. Solve for x by adding 5 to both sides:
x = 5 ± √3
Thus, the solutions are:
x = 5 + √3 and x = 5 – √3.
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Simplify this solution The Deep Dive
To solve the equation \((x-5)^{2}=3\) using the square root property, we start by taking the square root of both sides. Remember to consider both the positive and negative roots: \[ x - 5 = \sqrt{3} \quad \text{or} \quad x - 5 = -\sqrt{3}. \] Now, we isolate \(x\) by adding 5 to both equations: 1. \( x = 5 + \sqrt{3} \) 2. \( x = 5 - \sqrt{3} \) So the solutions are \( x = 5 + \sqrt{3} \) and \( x = 5 - \sqrt{3} \). To express these solutions in decimal form: - \( 5 + \sqrt{3} \approx 6.732 \) - \( 5 - \sqrt{3} \approx 3.268 \) And there you have it—two elegant solutions to your equation!
