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The quadratic formula is a powerful tool for finding the real zeros of a polynomial, especially when factoring is not straightforward or possible. It is formulated as follows:
\[ x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}} \]
For any quadratic equation in the standard form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, the quadratic formula can be applied. This formula uses the coefficients of the polynomial to calculate the zeros. The term 'real zeros' refers to the x-values where the polynomial crosses the x-axis when graphed. These are the solutions to the equation that are not complex numbers. When using the quadratic formula, the discriminant, \( b^2 - 4ac \), determines the nature of the roots. If it is positive, there are two distinct real zeros; if it is zero, there is one real zero; if it is negative, the zeros are complex.

Polynomial coefficients are the numerical factors that precede the variable terms in a polynomial expression. In the quadratic equation \( ax^2 + bx + c = 0 \), the coefficients are \( a \), \( b \), and \( c \). Each coefficient has a specific role in defining the shape and position of the parabola when graphed.

• \( a \) is the leading coefficient, impacting how wide or narrow the parabola is and whether it opens upward (\( a > 0 \)) or downward (\( a < 0 \)).


• \( b \) influences the horizontal position of the vertex of the parabola.


• \( c \) is the constant term and determines the vertical position of the parabola's vertex.

In our example, the coefficients are 3, 5, and 2 for \( a \), \( b \), and \( c \) respectively. These coefficients directly affect the calculation of the roots using the quadratic formula.

Solving quadratic equations is a fundamental concept in algebra, and there are several methods to do so. One of the most reliable is to use the quadratic formula, which guarantees a solution for any quadratic equation, whether the roots are real or complex.
In the context of this exercise, we applied the quadratic formula as follows: after identifying the coefficients \( a \), \( b \), and \( c \), we substitute them into the formula, and after simplifying, we derive the possible values for \( x \), which are the solutions to the equation. These solutions might be two different real numbers, one repeated real number, or a pair of complex numbers, depending on the value of the discriminant. By solving \( ax^2 + bx + c = 0 \) using the quadratic formula, students can confidently find the zeros of the given polynomial, as demonstrated through our example where the solutions were \( x_1= -\frac{2}{3} \) and \( x_2 = -1 \).