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The discriminant is a key part of solving quadratic equations using the quadratic formula. It is the part of the quadratic formula that lies under the square root: \(b^2 - 4ac\). This piece has a specific purpose, revealing important properties of the quadratic equation you're working with.

• **Positive Discriminant**: If the discriminant is positive, there will be two real and different solutions for the equation.
• **Zero Discriminant**: If it is zero, there will be exactly one real solution (or a double root), meaning the graph of the equation touches the x-axis at one point.
• **Negative Discriminant**: If the discriminant is negative, the equation has no real solutions but instead two complex solutions.

In our problem, the discriminant was calculated as \(24\), which is positive. This means we know before even calculating that there are two distinct real solutions to our quadratic equation.

When you're solving quadratic equations using the quadratic formula, it's like following a recipe. The formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) always works for these types of equations, making it a versatile and reliable method.
First, ensure your quadratic equation is in the standard form \(ax^2 + bx + c = 0\). This step is crucial since the formula relies on the values of \(a\), \(b\), and \(c\). In our exercise, the equation \(3x^2 - 6x + 1 = 0\) directly fit into the formula format.
Next, it's about plugging the coefficients into the formula meticulously:

• **Substitute** the values of \(a\), \(b\), and \(c\) into the formula. Here, that's \(a = 3\), \(b = -6\), \(c = 1\).
• **Solve** the expression under the square root (which is the discriminant) to anticipate the type and number of solutions.
• **Calculate** the final value of \(x\) using simple arithmetic, applying the appropriate operations as indicated by the \(\pm\) sign.

This step-by-step process ensures accuracy and clarity, leading you confidently to the solutions.

The final step in using the quadratic formula is finding the actual solutions, or roots, of the equation. After calculating the discriminant and affirming it's positive, the solutions can be extracted from the formula.
In our solved exercise, the discriminant \(\sqrt{24}\) appears in our expressions, forming two separate outcomes due to the \(\pm\) symbol. So, our solution becomes:

• **First Solution**: \(x = 1 + \frac{\sqrt{24}}{6}\)
• **Second Solution**: \(x = 1 - \frac{\sqrt{24}}{6}\)

These steps will yield numeric results once you compute \(\sqrt{24}/6\). Breaking things down further, \(\sqrt{24}\) simplifies to \(2\sqrt{6}\), giving you clearer numerical values when substituted back.
Understanding that each quadratic equation potentially leads to two roots is critical. These roots are the points where the equation's graph, usually a parabola, intersects or would intersect the x-axis.