If we express 3x^2 - 6x + 2 + x^2 - 2x + 7 in the form a(x - h)^2 + k, then what is a + h + k?
(This will be a step by step explanation instead of using completing the square formulas/shortcuts)
First we must simplify the original expression.
\(4x^2 - 8x + 9\) this expression looks cleaner.
Since we have to complete the square, we know \(a\), the coefficient of \(x^2\) has to be 4.
If we open the parenthesis of \((x - h)^2\), we get \(x^2 + h^2 - 2hx\).
Then since we are multiplying 4 to the expression, we have \(4x^2 + 4h^2 - 8hx\).
\(h\) is a constant. if \(-8hx\) is the only value with \(x^1\), then \(-8h = -8\) in the original expression. Thus \(h = 1\).
The new expression \(4x^2 - 8x + 4\) has a difference of a positive 5 than the original expression \(4x^2 - 8x + 9\).
Thus \(k = 5\).
Adding up our values, we have a + h + k = 4 + 1 + 5 = 10.