The function \(f\) is defined by \(f(x) = x^2 - 4x + 8\) for \(x \in \mathbb{R}\). Express \(x^2 - 4x + 8\) in the form \((x-a)^2 + b\).
Solution
To express \(x^2 - 4x + 8\) in the form \((x-a)^2 + b\), we complete the square:
1. Start with the expression: \(x^2 - 4x + 8\).
2. Take the coefficient of \(x\), which is \(-4\), halve it to get \(-2\), and square it to get \(4\).
3. Add and subtract \(4\) inside the expression:
\(x^2 - 4x + 4 - 4 + 8\)
4. Rewrite as a perfect square and simplify:
\((x-2)^2 + 4\)
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