Question

Solve the following quadratic equation using the perfect square trinomial pattern: x^2 + 10x + 25 = 0

Answer 1

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.