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When faced with a quadratic equation, one of the most versatile methods to find its roots is by using the quadratic formula. This formula provides a solution to any quadratic equation of the form \(ax^2 + bx + c = 0\). The roots of the equation can be given by the formula:

\[x = \frac{-b \[b^2 - 4ac}}{2a}\]
The letters \(a\), \(b\), and \(c\) represent the coefficients of the equation, where \(a\) is the coefficient of the quadratic term, \(b\) is the coefficient of the linear term, and \(c\) is the constant term. In the context of the exercise:

• \(a\) corresponds to 3 (the coefficient of \(y^2\)),
• \(b\) is 7 (the coefficient of \(y\)),
• \(c\) is -24 (the constant term).

By substituting these values into the quadratic formula, we can find the values of \(y\) which satisfy the original quadratic equation.

To solve quadratic equations efficiently, we often need to combine like terms to write the equation in its standard form, \(ax^2 + bx + c = 0\). Combining like terms means adding or subtracting terms with the same variable and the same exponent. In the exercise, we start with the equation \(3y^2 - 7y - 24 = 0\), which is derived from simplifying the original equation.

This simplification involves distributing the coefficient across terms within parentheses, like multiplying 3 by (\(y^2\)) and (-8), and then carrying over like terms to one side of the equation. By doing this, we aligned the equation to a structure where the quadratic formula could be readily applied. Understanding how to combine like terms is crucial, as it transforms the equation into a workable form for further analysis.

The discriminant is a particular part of the quadratic formula denoted by \(\Delta\), and it provides critical information about the nature of the roots of a quadratic equation. The discriminant is found under the square root in the quadratic formula, expressed as \(b^2 - 4ac\). The value of the discriminant determines the number and type of roots:

• If \(\Delta > 0\), there are two real and distinct solutions.
• If \(\Delta = 0\), there is exactly one real solution.
• If \(\Delta < 0\), there are no real solutions; instead, there are two complex solutions.

In the exercise, the discriminant value was calculated as 337, which is greater than zero. Therefore, we could confirm there would be two unique real solutions for the value of \(y\). These solutions are obtained by applying the ± symbol of the quadratic formula, generating the two real roots of the quadratic equation.