Question
\begin{tabular}{l} Solve \\ \( (x-5)^{2}=4 \) \\ \hline 80\end{tabular}
Ask by Tyler Hall. in the United States
Jan 24,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The solutions are \(x=3\) and \(x=7\).
Solution
Solve the equation \( (x-5)^{2}=4 \).
Solve the quadratic equation by following steps:
- step0: Solve using square roots:
\(\left(x-5\right)^{2}=4\)
- step1: Simplify the expression:
\(x-5=\pm \sqrt{4}\)
- step2: Simplify the expression:
\(x-5=\pm 2\)
- step3: Separate into possible cases:
\(\begin{align}&x-5=2\\&x-5=-2\end{align}\)
- step4: Solve the equation:
\(\begin{align}&x=7\\&x=3\end{align}\)
- step5: Rewrite:
\(x_{1}=3,x_{2}=7\)
The solutions to the equation \((x-5)^{2}=4\) are \(x=3\) and \(x=7\).
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Simplify this solution Bonus Knowledge
To solve the equation \( (x-5)^{2}=4 \), we first take the square root of both sides. This gives us two possible equations: \( x-5=2 \) and \( x-5=-2 \). Solving these, we have \( x=7 \) from the first equation and \( x=3 \) from the second equation. Thus, the solutions to the equation are \( x=7 \) and \( x=3 \). Now, let's double-check our solutions by plugging them back into the original equation. For \( x=7 \), \( (7-5)^{2} = 2^{2} = 4 \), and for \( x=3 \), \( (3-5)^{2} = (-2)^{2} = 4 \) as well. Both solutions check out!
