Express \(x^2 + 6x + 5\) in the form \((x + a)^2 + b\), where \(a\) and \(b\) are constants.
Solution
To express \(x^2 + 6x + 5\) in the form \((x + a)^2 + b\), we complete the square:
Start with the expression: \(x^2 + 6x + 5\).
1. Take half of the coefficient of \(x\), which is 6, giving 3, and square it: \(3^2 = 9\).
2. Add and subtract this square inside the expression:
\(x^2 + 6x + 9 - 9 + 5\).
3. Rewrite the expression as a perfect square and a constant:
\((x + 3)^2 - 4\).
Thus, \(a = 3\) and \(b = -4\).
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