To solve the quadratic equation \( 49x^2 + 392x + 784 = 0 \) using the perfect square trinomial pattern, we first rewrite the equation in the standard form:
\[ 49x^2 + 392x + 784 = 0 \]
Next, we can factor the left-hand side. We notice that the equation has coefficients that suggest a perfect square trinomial:
The expression can be rewritten as:
\[ (7x + 28)^2 = 0 \]
To verify this, we can expand \( (7x + 28)^2 \):
\[ (7x + 28)(7x + 28) = 49x^2 + 392x + 784 \]
This is correct, so we set the equation equal to zero:
\[ (7x + 28)^2 = 0 \]
Now, we take the square root of both sides:
\[ 7x + 28 = 0 \]
Next, we solve for \( x \):
\[ 7x = -28 \] \[ x = -4 \]
So, the solution to the equation \( 49x^2 = -392x - 784 \) is:
\[ \boxed{-4} \]