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A quadratic equation is a polynomial equation of the form \(ax^2 + bx + c = 0\). The solutions to a quadratic equation can often be found using various methods like factoring, the quadratic formula, or completing the square. In this exercise, we focus on completing the square, which involves turning a standard quadratic equation into a perfect square trinomial. Understanding the standard form—where \(a = 1\), \(b = 12\), and \(c = 32\) for our equation \(x^2 + 12x + 32 = 0\)—is crucial. All these values play a key role in the steps we perform next.

Completing the square helps us convert a quadratic expression into a perfect square trinomial. Here's how it works with our problem \(x^2 + 12x + 32 = 0\):
Step 1: We start by moving the constant term to the other side \(x^2 + 12x = -32\).
Step 2: We find the value that makes the quadratic a perfect square. Take half the coefficient of the linear term (which is 12), and square it. That gives us \(6^2 = 36\).
Add this value to both sides, transforming the equation into \(x^2 + 12x + 36 = 4\).
Now it’s clear that the left side is a perfect square trinomial, and can be written as \((x + 6)^2 = 4\).
As a perfect square trinomial, it’s easy to solve by taking the square root of both sides.

Once we have a perfect square trinomial, solving the equation becomes straightforward. From our example, we solved by turning \((x + 6)^2 = 4\) into two separate linear equations since \(\text{sqrt}(4) = \text{±2}\). This gives us:
1. \(x + 6 = 2\) - Solve for \(x\), yielding \(x = -4\)
2. \(x + 6 = -2\) - Solve for \(x\), yielding \(x = -8\)

By isolating \(x\) on one side in each case, we identify the solutions to the original quadratic equation. Therefore, the solutions to the initial problem \(x^2 + 12x + 32 = 0\) are \(x = -4\) and \(x = -8\). Understanding these steps highlights the effectiveness of completing the square as an alternative method to solving quadratic equations.