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The process of solving quadratic equations is fundamental in algebra. When you're dealing with an equation like ax^2 + bx + c = 0, you're working with a standard form of a quadratic equation. The solution requires finding the value of x that makes the equation true.

One reliable method for solving quadratics is using the quadratic formula: \( x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} \). This formula provides a systematic way to find the roots of any quadratic equation. The a, b, and c in the formula correspond to the coefficients in the equation you are trying to solve. Always remember to express the equation in its standard form before applying the formula.

Let's consider our example, \(4s^2 + 12s - 3 = 0\), the coefficients are \(a = 4\), \(b = 12\), and \(c = -3\). By substituting these values into the quadratic formula, you will find the values of s that solve the equation. This is exactly what the step-by-step solution has shown, guiding you towards understanding how to apply the quadratic formula to real problems.

The discriminant in a quadratic equation is a powerful tool that gives us valuable insights about the nature of roots without actually solving the equation. It is given by the formula: \(D = b^2 - 4ac\). This value determines whether the quadratic has real or complex roots and whether they are distinct or repeated.

If the discriminant is positive, the quadratic has two distinct real roots. A discriminant of zero indicates that there is exactly one real root (also called a repeated root). However, if the discriminant is negative, the equation has no real roots; instead, it has two complex roots. In the example \(4s^2 + 12s - 3 = 0\), the calculated discriminant of 192 signifies that the equation has two distinct real roots. We found this by substituting the coefficients into the discriminant formula: \( 12^2 - 4 \times 4 \times (-3) = 144 + 48 \).

When we refer to the 'roots' of a quadratic equation, we are talking about the solutions to the equation; basically, they are the values of x that satisfy the equation \(ax^2 + bx + c = 0\). The quadratic formula gives us a direct path to these roots.

In the exercise, \(-12 \pm \sqrt{192} / 2 \times 4 = -12 \pm (8\sqrt{3}) / 8 = -1.5 \pm \sqrt{3}\), we see that the roots are obtained through a specific process: simplifying the quadratic formula after putting in all the necessary values. This process will always give us the most accurate possible values for the roots. By understanding the nature of these roots – whether they are real or complex, and distinct or repeated – we are more equipped to handle different kinds of quadratic equations. Remember, finding the roots is akin to unlocking the equation; once you have them, you know all the possible solutions!