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The quadratic formula is a vital tool for solving quadratic equations. These equations take the form:\[ ax^2 + bx + c = 0 \]where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\). The quadratic formula allows you to find the roots (or zeros) of the equation, and it is given by:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]Using this formula, you can ensure that you account for complex and real solutions. The part under the square root, \(b^2 - 4ac\), is called the discriminant. Solving quadratic equations using this formula is efficient and clear, especially because it provides a systematic way to arrive at the solutions without needing to factor the quadratic expression manually.

In the context of quadratic equations, the discriminant plays a crucial role in determining the nature of the roots. It is the expression found under the square root in the quadratic formula: \(b^2 - 4ac\). This value can tell us a lot about the equation:

• If the discriminant \(\Delta > 0\), the quadratic equation has two distinct real roots.
• If \(\Delta = 0\), there is exactly one real root, which means the graph touches the x-axis at a single point.
• If \(\Delta < 0\), the equation has no real roots, and instead, there are two complex conjugate roots.

In the given exercise, the discriminant is \(625\), which is positive, indicating two distinct real roots. This ensures that the parabola will intersect the x-axis in two distinct points, aligning with the solutions found.

The vertex of a quadratic function is an essential feature of its graph, as it represents the point where the function changes direction. For the standard form \(ax^2 + bx + c\) of a quadratic equation, the x-coordinate of the vertex is given by:\[ x = -\frac{b}{2a} \]This formula gives a quick way to find where the vertex lies on the x-axis. Once the x-value is determined, substitute it back into the quadratic function to find the corresponding y-coordinate. This point \((x, y)\) is the vertex.

• If the parabola opens upwards (\(a > 0\)), the vertex is the minimum point.
• If the parabola opens downwards (\(a < 0\)), the vertex is the maximum point.

In this particular exercise, with \(a = 6\), and the vertex was found at \(-\frac{7}{12}\) for the x-coordinate, leading to a minimum value point, due to the parabola opening upwards.

Graphing parabolas involves understanding the shape and key points of the quadratic function's graph. The fundamental characteristics of a parabola include its vertex, direction of opening, and roots or zeros:

• Vertex: The turning point of the parabola, which we calculated as \((-\frac{7}{12}, -\frac{625}{24})\) in our example.
• Zeros (Roots): The x-values where the parabola intersects the x-axis, calculated using the quadratic formula.
• Opening Direction: Determined by the sign of \(a\); if positive, the parabola opens upwards, as seen here with \(a = 6\).

To sketch the graph:1. Plot the vertex.2. Mark the x-intercepts (the zeros found at \(\frac{3}{2}\) and \(-\frac{8}{3}\)).3. Ensure that the parabola is symmetric around the vertical line that passes through its vertex.This approach provides clear guidance on representing the quadratic function graphically.