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The quadratic formula is a powerful tool for solving quadratic equations that are presented in the standard form, \(ax^2 + bx + c = 0\). It offers a straightforward method to find solutions, or "roots," of these equations by substituting the constants into the formula:

• \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

In using this formula, we solve for \(x\) by plugging in the values of \(a\), \(b\), and \(c\), which are constants from the quadratic equation.
For example, in the exercise given, \(a = 3\), \(b = -2\), and \(c = -11\). These correspond directly to the structure of the quadratic formula. The formula helps isolate \(x\) and determine its actual numerical values or zeros, if they exist.
Understanding the quadratic formula is crucial because it allows us to work with any quadratic equation, even when factoring is difficult or impossible.

The discriminant is a part of the quadratic formula, specifically the expression under the square root: \(b^2 - 4ac\). This part of the equation plays a vital role in telling us how many and what type of solutions the quadratic equation will have.
Here is what the discriminant can reveal:

• If \(b^2 - 4ac > 0\), the equation has two distinct real solutions.
• If \(b^2 - 4ac = 0\), there is exactly one real solution (also known as a repeated root).
• If \(b^2 - 4ac < 0\), the quadratic equation has no real solutions, only complex ones.

In the solution given, we calculated the discriminant as 136, which is greater than zero. This confirms that there are two distinct real solutions for the quadratic equation \(3x^2 - 2x - 11 = 0\).

Zeros of a function, in a mathematical context, are the x-values that make the output (y-value) of the function equal to zero. For a quadratic function, such as \(y = 3x^2 - 2x - 11\), the zeros or roots are found using the quadratic formula.
The solutions \(x_1\) and \(x_2\) (found using the formula) correspond directly to these zeros. In the example from the exercise, we have two zeros:

• \(x_1 \approx 1.354\)
• \(x_2 \approx -2.688\)

These zeros indicate the points where the graph of the quadratic function crosses the x-axis.
Understanding zeros is essential for graphing and analyzing quadratic functions, as they provide crucial insights into the shape and position of the parabola that represents the function on a coordinate plane.