Question
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Answer
The solutions are \( x=2 \) and \( x=3 \).
Solution
Solve the equation \( (x-2)(x-2)(3-x)=0 \).
Solve the equation by following steps:
- step0: Solve for \(x\):
\(\left(x-2\right)\left(x-2\right)\left(3-x\right)=0\)
- step1: Multiply the first two terms:
\(\left(x-2\right)^{2}\left(3-x\right)=0\)
- step2: Separate into possible cases:
\(\begin{align}&\left(x-2\right)^{2}=0\\&3-x=0\end{align}\)
- step3: Solve the equation:
\(\begin{align}&x=2\\&x=3\end{align}\)
- step4: Rewrite:
\(x_{1}=2,x_{2}=3\)
The solutions to the equation \( (x-2)(x-2)(3-x)=0 \) are \( x=2 \) and \( x=3 \).
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Simplify this solution Extra Insights
To solve \( (x-2)(x-2)(3-x)=0 \), we can set each factor to zero. First, since \( (x-2) \) appears twice, we have \( x-2=0 \) leading to a double root at \( x=2 \). Next, from \( (3-x)=0 \), we find \( x=3 \). Thus, the solutions are \( x=2 \) (with multiplicity 2) and \( x=3 \). Now the equation can be expressed as \( x = 2 \) (twice) and \( x = 3 \). Happy solving!
