To solve the quadratic equation \(49x^2 = -392x - 784\) using the perfect square trinomial pattern, we first rearrange the equation into the standard form of a quadratic:
\[ 49x^2 + 392x + 784 = 0. \]
Next, we can factor out the common coefficient from the left side:
\[ 49(x^2 + 8x + 16) = 0. \]
Now, inside the parentheses, we recognize that \(x^2 + 8x + 16\) is a perfect square trinomial, which can be factored as:
\[ (x + 4)^2. \]
Thus, our equation simplifies to:
\[ 49(x + 4)^2 = 0. \]
To solve for \(x\), we set the equation inside the parentheses equal to zero:
\[ (x + 4)^2 = 0. \]
Taking the square root of both sides, we get:
\[ x + 4 = 0, \]
which leads to:
\[ x = -4. \]
Thus, the solution to the equation is:
\[ \boxed{-4}. \]
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