Solve the quadratic equation x^2 + 6x + 9 = 0 for x.

To solve the quadratic equation ( x^2 + 6x + 9 = 0 ), we can recognize that this equation can be factored. The equation represents a perfect square trinomial, which can be expressed as:

[ (x + 3)(x + 3) = 0

]

This factors into ( (x + 3)^2 = 0 ). Setting the factored form equal to zero gives us:

[ x + 3 = 0 ]

Solving for ( x ), we subtract 3 from both sides:

[ x = -3 ]

Thus, the solution to the equation ( x^2 + 6x + 9 = 0 ) is ( x = -3 ). This also means that the quadratic has a double root at ( x = -3 ), indicating that the curve touches the x-axis at this point and does not cross it.

Recognizing this type of factorization and solving via the roots method shows how the perfect square form helps isolate the variable and find the solution efficiently.