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The quadratic formula is a powerful tool used to find the solutions of quadratic equations. Quadratic equations are in the form of \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants. This formula allows us to find the "roots" or "zeros" of the function, which are the values of \( x \) that make the entire quadratic equation equal to zero.
The quadratic formula is given by:\[x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\]Here's how it works:

• The symbols \( + \) and \( - \) indicate that there are typically two solutions — one for addition and one for subtraction.
• The "\( a \)" is the coefficient of \( x^2 \), "\( b \)" is the coefficient of \( x \), and "\( c \)" is the constant term.

By substituting the values of \( a \), \( b \), and \( c \) from a given quadratic equation into this formula, we can swiftly calculate the zeros.
In our example, we had \( a = 2 \), \( b = -7 \), and \( c = -30 \), leading to roots \( x = 5 \) and \( x = -3 \). This systematic approach saves time, particularly when factoring seems complex.

The discriminant is a crucial part of the quadratic formula, helping determine the nature and number of the solutions. It is found under the square root sign and is represented as \( b^2 - 4ac \). Here’s what the discriminant tells us:

• If \( b^2 - 4ac > 0 \), there are two distinct real solutions. The parabola will intersect the x-axis at two points.
• If \( b^2 - 4ac = 0 \), there is exactly one real solution. The parabola touches the x-axis at only one point, also known as a double root.
• If \( b^2 - 4ac < 0 \), there are no real solutions. The parabola does not intersect the x-axis, as the roots are complex numbers.

In this exercise, the discriminant was \( 289 \), which is positive. Therefore, our quadratic equation has two distinct real solutions. Understanding the discriminant gives us insight into the resultant graph of the quadratic function even before actually solving it.

Zeros of a function, also known as roots or solutions, are the x-values that make a function equal to zero. For a quadratic function like \( f(x)=2x^2-7x-30 \), finding the zeros involves solving the equation \( 2x^2 - 7x - 30 = 0 \).
Zeros are essential because they tell us where the parabola crosses the x-axis. They help in graphing the function and understanding its behavior. These values may appear as:

• Real and distinct: The parabola crosses the x-axis at two separate points.
• Real and repeated: The parabola touches but does not cross the x-axis, indicating a vertex on the axis.
• Complex: The parabola does not touch or intersect the x-axis at any point.

From our earlier solution, the zeros were \( x = 5 \) and \( x = -3 \), meaning the graph crosses the x-axis at these two points. Recognizing zeros is crucial as they provide solutions to various real-world problems modeled by quadratic equations.