Express \(x^2 + 6x + 2\) in the form \((x + a)^2 + b\), where \(a\) and \(b\) are constants.
Solution
To express \(x^2 + 6x + 2\) in the form \((x + a)^2 + b\), we complete the square:
1. Start with the expression: \(x^2 + 6x + 2\).
2. Take half of the coefficient of \(x\), which is 6, giving 3, and square it: \(3^2 = 9\).
3. Add and subtract this square inside the expression: \(x^2 + 6x + 9 - 9 + 2\).
4. Rewrite as a perfect square: \((x + 3)^2 - 9 + 2\).
5. Simplify the constants: \((x + 3)^2 - 7\).
Thus, \(a = 3\) and \(b = -7\).
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