To solve the quadratic equation \( \frac{1}{4}x^2 + 2x + 4 = 0 \) using the perfect square trinomial pattern, we first want to express it in a form that allows us to factor it as a perfect square.
- Start by eliminating the fraction. We can multiply the entire equation by 4 to make calculations easier:
\[ 4 \left( \frac{1}{4}x^2 + 2x + 4 \right) = 4 \cdot 0 \] \[ x^2 + 8x + 16 = 0 \]
- Now the equation is \( x^2 + 8x + 16 = 0 \). Notice that \( x^2 + 8x + 16 \) is a perfect square trinomial. It can be factored as:
\[ (x + 4)^2 = 0 \]
- Now, solve for \( x \) by taking the square root of both sides:
\[ x + 4 = 0 \]
- Subtract 4 from both sides:
\[ x = -4 \]
Thus, the solution to the equation \( \frac{1}{4}x^2 + 2x + 4 = 0 \) is:
\[ \boxed{-4} \]
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