To express \(2x^2 - 8x + 14\) in the form \(2[(x-a)^2 + b]\), we follow these steps:
1. Start with the expression: \(2x^2 - 8x + 14\).
2. Factor out the 2 from the first two terms: \(2(x^2 - 4x) + 14\).
3. Complete the square inside the parentheses:
- Take half of the coefficient of \(x\), which is \(-4\), giving \(-2\).
- Square it to get \(4\).
4. Add and subtract this square inside the parentheses:
\(2[(x^2 - 4x + 4) - 4] + 14\).
5. Simplify the expression:
\(2[(x-2)^2 - 4] + 14\).
6. Distribute the 2 and simplify:
\(2(x-2)^2 - 8 + 14\).
7. Combine the constants:
\(2(x-2)^2 + 6\).
8. Rewrite in the desired form:
\(2[(x-2)^2 + 3]\).
