To convert the quadratic function \( f(x) = x^2 + 12x + 6 \) into vertex form, we can use the method of completing the square.
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Start with the original equation: \[ f(x) = x^2 + 12x + 6 \]
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Focus on the quadratic and linear terms \( x^2 + 12x \). To complete the square, take half of the coefficient of \( x \) (which is 12), square it, and add and subtract this value inside the function. Half of 12 is 6, and \( 6^2 = 36 \).
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Rewrite the function: \[ f(x) = (x^2 + 12x + 36) - 36 + 6 \] \[ f(x) = (x + 6)^2 - 30 \]
Now, we have the function in vertex form \( f(x) = (x + 6)^2 - 30 \).
From this, we can see that:
- \( h = -6 \) (since the vertex form is \( (x - h)^2 + k \))
- \( k = -30 \)
Thus, the values for \( h \) and \( k \) are:
\[ \text{h} = -6, \text{ k} = -30 \]
So the correct answer is: h = -6, k = -30.