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Answer
The solutions are \( w=0 \) and \( w=13 \).
Solution
Solve the equation \( (w-4)^{2}=16+5w \).
Solve the quadratic equation by following steps:
- step0: Solve by factoring:
\(\left(w-4\right)^{2}=16+5w\)
- step1: Expand the expression:
\(w^{2}-8w+16=16+5w\)
- step2: Move the expression to the left side:
\(w^{2}-13w=0\)
- step3: Factor the expression:
\(w\left(w-13\right)=0\)
- step4: Separate into possible cases:
\(\begin{align}&w-13=0\\&w=0\end{align}\)
- step5: Solve the equation:
\(\begin{align}&w=13\\&w=0\end{align}\)
- step6: Rewrite:
\(w_{1}=0,w_{2}=13\)
The solutions to the equation \((w-4)^{2}=16+5w\) are \(w=0\) and \(w=13\).
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Simplify this solution Beyond the Answer
To solve the equation \((w-4)^{2}=16+5 w\), let's start by expanding the left side: \[ (w-4)^{2} = w^{2} - 8w + 16. \] Now, we can rewrite the equation: \[ w^{2} - 8w + 16 = 16 + 5w. \] Next, subtract \(16 + 5w\) from both sides: \[ w^{2} - 8w + 16 - 16 - 5w = 0. \] This simplifies to: \[ w^{2} - 13w = 0. \] Now, factor out \(w\): \[ w(w - 13) = 0. \] Setting each factor to zero gives us: \[ w = 0 \quad \text{or} \quad w - 13 = 0 \Rightarrow w = 13. \] Thus, the solutions to the equation are: \[ w = 0 \quad \text{or} \quad w = 13. \] **Final Answer: \( w = 0 \) or \( w = 13 \)**
