To express the given quadratic expression \(2x^2 - 12x + 23\) in the form \(a(x + b)^2 + c\), we'll complete the square:
1. First, factor out the leading coefficient, which is 2:
\(2x^2 - 12x + 23 = 2(x^2 - 6x) + 23\)
2. Complete the square inside the parentheses by adding and subtracting \((\text{coefficient of } x)^2\):
\(2(x^2 - 6x + 9) - 2(9) + 23\)
3. Simplify the expression:
\(2(x - 3)^2 - 2(9) + 23\)
4. Further simplify:
\(2(x - 3)^2 - 18 + 23\)
5. Combine constants:
\(2(x - 3)^2 + 5\)
So, in the form \(a(x + b)^2 + c\), where \(a = 2\), \(b = -3\), and \(c = 5\), the expression is \(2(x - 3)^2 + 5\).
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