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Understanding the standard form of a quadratic equation is crucial for solving quadratic equations. It serves as the starting point for many solution methods. The general standard form is given by \( ax^2 + bx + c = 0 \), where \(a\), \(b\), and \(c\) are constants, and \(x\) is the variable we aim to solve for. Moreover, \(a \eq 0\), as if \(a=0\), the equation would become linear, not quadratic.

When faced with a quadratic equation not in standard form, such as \( u^2 = -5u - 2 \), the initial step is to rearrange it to resemble the standard form. This involves moving all terms to one side of the equation, leading to \( u^2 + 5u + 2 = 0 \). Now in the correct format, we can readily identify the coefficients \(a=1\), \(b=5\), and \(c=2\), which are essential for applying the quadratic formula and finding the roots of the equation.

The process of solving quadratic equations often commences after converting them into standard form. With our example \(u^2 + 5u + 2 = 0\), one of the most robust methods for finding the solutions, or roots, is to use the Quadratic Formula. This remarkable formula, \( u = \frac{-b \pm \sqrt{b^2-4ac}}{2a} \), is designed to solve any quadratic equation, regardless of the value of the coefficients.

By substituting our coefficients into the formula, \( u = \frac{-5 \pm \sqrt{5^2 - 4 \cdot 1 \cdot 2}}{2 \cdot 1} \), we derive the possible solutions for \(u\). The \(\pm\) symbol indicates two solutions, one involving addition and the other, subtraction. In layman’s terms, we are simply finding two numbers that, when replaced for \(u\), make the entire equation equal zero - these numbers are the 'roots' of the equation.

The term roots refers to the solutions of the equation where it equals zero. They are the values of \(x\) that make the quadratic expression \( ax^2 + bx + c \) equal to zero. Using the Quadratic Formula from our example, we have \( u = \frac{-5 \pm \sqrt{5^2 - 4 \cdot 1 \cdot 2}}{2 \cdot 1} \), which simplifies to \( u = \frac{-5 \pm \sqrt{17}}{2} \).

By carrying out the operations within the formula, we establish two distinct roots: \( u = \frac{-5 + \sqrt{17}}{2} \) and \( u = \frac{-5 - \sqrt{17}}{2} \), depending on whether we add or subtract the square root portion. These solutions are critical as they answer the fundamental question: Which values can \(u\) take on such that the original quadratic equation holds true? Understanding the process deeply not only aids in solving textbook problems but also underpins many practical and theoretical applications in various fields, including physics, engineering, and economics.