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Factoring quadratic equations can sometimes seem daunting, but it's a straightforward process when broken down. When you have a quadratic like \(x^2 - 12x + 36 = 0\), your goal is to express it as a product of two binomials. The reason we do this is that finding out where this product equals zero will tell us the roots or solutions of the quadratic. Here, we factor it into \((x-6)(x-6) = 0\).
This factorization tells us that the expression becomes zero when \(x = 6\). That's because any number multiplied by zero equals zero. You look for two numbers that multiply to the last term (36, in this case) and add up to the middle coefficient (-12). Once you recognize that both terms are \(-6\) and \(-6\), you've found your solution. Factorization simplifies the process of solving quadratics dramatically by turning a square into a product of 2 linear expressions.

The standard form of a quadratic is usually written as \(ax^2 + bx + c = 0\). This form is incredibly useful because it allows us to apply various solution techniques, such as factoring, completing the square, or using the quadratic formula.
For the given problem, the equation \(x^2 = 12x - 36\) needed to be rearranged. By subtracting \(12x\) from both sides and adding \(36\), we rewrote it in the standard form as \(x^2 - 12x + 36 = 0\).
This transformation is crucial for consistency in solving different quadratic equations, since the standard form highlights all necessary terms for identification. Doing this step first ensures that no term is neglected in the process.

After solving the quadratic equation, verifying the roots ensures accuracy. Suppose our solution to the equation was \(x = 6\). To verify, substitute \(x = 6\) back into the original equation: \(x^2 = 12x - 36\).
This gives \(36 = 72 - 36\), which simplifies into \(36 = 36\).
This verification step is simple and gives peace of mind that the solution works, both algebraically and numerically. In cases where you are not confident, always double-check by plugging the solution back into the equation to see if the left-hand side equals the right-hand side.

Apart from the algebraic route, we can also use graphical methods to solve quadratics. This involves graphing the quadratic equation and finding where it intersects the x-axis, known as the x-intercepts. For our equation \(x^2 - 12x + 36 = 0\), we can plot the curve of \(y = x^2 - 12x + 36\).
Upon graphing, you will notice the parabola touches the x-axis at a single point \(x = 6\).
This point of intersection represents the root of the quadratic equation. With the help of graphing utilities or graph paper, this method confirms our algebraic solution visually. This method acts as a bridge connecting geometric intuition with algebraic precision. Using graphs can be a powerful tool to understand the behavior of quadratics beyond specific solutions.