solve `x^(2)+6x+9=`

Answer

To solve the quadratic equation \( x^2 + 6x + 9 = 0 \), we can follow these steps:

### Step 1: Identify the quadratic equation
The given equation is:
\[ x^2 + 6x + 9 = 0 \]

### Step 2: Factor the quadratic expression
We can factor the quadratic expression. We need to find two numbers that multiply to \( 9 \) (the constant term) and add up to \( 6 \) (the coefficient of \( x \)). The numbers \( 3 \) and \( 3 \) satisfy this condition since:
\[ 3 + 3 = 6 \quad \text{and} \quad 3 \times 3 = 9 \]
Thus, we can rewrite the equation as:
\[ x^2 + 3x + 3x + 9 = 0 \]

### Step 3: Group the terms
Now, we can group the terms:
\[ (x^2 + 3x) + (3x + 9) = 0 \]

### Step 4: Factor by grouping
Next, we factor out the common terms:
\[ x(x + 3) + 3(x + 3) = 0 \]
This can be factored further:
\[ (x + 3)(x + 3) = 0 \]
or simply:
\[ (x + 3)^2 = 0 \]

### Step 5: Solve for \( x \)
Now, we set the factored expression equal to zero:
\[ (x + 3)^2 = 0 \]
Taking the square root of both sides gives:
\[ x + 3 = 0 \]
Thus, solving for \( x \):
\[ x = -3 \]

### Final Answer
The solution to the equation \( x^2 + 6x + 9 = 0 \) is:
\[ \boxed{-3} \]

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