To solve the quadratic equation \( \frac{1}{4} x^2 + 2x + 4 = 0 \) using the perfect square trinomial pattern, we will first rewrite the equation in a form that makes it easier to identify a perfect square.
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Multiply the entire equation by 4 to eliminate the fraction:
\[ x^2 + 8x + 16 = 0 \]
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Notice that \( x^2 + 8x + 16 \) can be rewritten as a perfect square:
\[ (x + 4)^2 = 0 \]
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Set the perfect square equal to zero:
\[ (x + 4)^2 = 0 \]
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Solve for \( x \) by taking the square root of both sides:
\[ x + 4 = 0 \]
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Finally, isolate \( x \):
\[ x = -4 \]
Thus, the solution to the equation \( \frac{1}{4} x^2 + 2x + 4 = 0 \) is
\[ \boxed{-4} \]
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