To solve the quadratic equation \( x^2 + 10x + 25 = 0 \) using the perfect square trinomial pattern, we first recognize that the left side of the equation can be rewritten as a square of a binomial.
The given expression can be rearranged as follows:
\[ x^2 + 10x + 25 = (x + 5)^2 \]
This is because: \[ (x + 5)(x + 5) = x^2 + 5x + 5x + 25 = x^2 + 10x + 25 \]
Now, we can substitute this back into the equation:
\[ (x + 5)^2 = 0 \]
Next, we take the square root of both sides:
\[ x + 5 = 0 \]
Now, we can solve for \( x \) by isolating it:
\[ x = -5 \]
Thus, the solution to the quadratic equation \( x^2 + 10x + 25 = 0 \) is:
\[ \boxed{-5} \]
We can verify this by substituting \( x = -5 \) back into the original equation:
\[ (-5)^2 + 10(-5) + 25 = 25 - 50 + 25 = 0 \]
Since this holds true, our solution \( x = -5 \) is correct.