Completing the square is a powerful technique used to solve quadratic equations that cannot be easily factored. This method transforms a quadratic equation into the form \(x + a\)^2 = c, allowing the use of the square root property to find solutions. To illustrate this process, consider the equation \(x^2 + 6x = -7\). The first step is to rearrange the equation into the standard form \(x^2 + bx = c\), ensuring that the leading coefficient is 1 and the constant is isolated on one side.

In this case, the leading coefficient is already 1, and we can proceed by adding \(\frac{b}{2}^2\) to both sides. Here, \(b\) is 6, so we calculate \(\frac{6}{2} = 3\) and then square it to get 9. Adding 9 to both sides gives us:

\(x^2 + 6x + 9 = -7 + 9\)

This simplifies to:

\(x^2 + 6x + 9 = 2\)

Next, we factor the left side, which becomes \((x + 3)^2 = 2\). Now that the equation is in the desired form, we can apply the square root property. Taking the square root of both sides yields:

\(x + 3 = \pm \sqrt{2}\)

To isolate \(x\), we subtract 3 from both sides, resulting in:

\(x = -3 \pm \sqrt{2}\)

This gives us the final solutions: \(x = -3 + \sqrt{2}\) and \(x = -3 - \sqrt{2}\).

Completing the square can be applied to any quadratic equation, but it is particularly effective when the leading coefficient is 1 and \(b\) is even. For example, consider the equation \(x^2 + 8x + 1 = 0\). First, we move the constant to the other side:

\(x^2 + 8x = -1\)

Next, we add \(\frac{b}{2}^2\) to both sides. Here, \(b\) is 8, so \(\frac{8}{2} = 4\) and squaring it gives us 16. Adding 16 to both sides results in:

\(x^2 + 8x + 16 = -1 + 16\)

This simplifies to:

\(x^2 + 8x + 16 = 15\)

Factoring the left side yields \((x + 4)^2 = 15\). Applying the square root property, we have:

\(x + 4 = \pm \sqrt{15}\)

Isolating \(x\) gives:

\(x = -4 \pm \sqrt{15}\)

Thus, the solutions are \(x = -4 + \sqrt{15}\) and \(x = -4 - \sqrt{15}\).

In summary, completing the square is a systematic method that allows for the solution of quadratic equations by transforming them into a form suitable for applying the square root property. This technique is versatile and can be used in various scenarios, making it an essential tool in algebra.