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The quadratic formula is a pivotal tool for solving quadratic equations—which are equations of the form ax²+bx+c=0. The formula itself is expressed as \( x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} \). To apply it effectively, identify the coefficients corresponding to a (the quadratic term), b (the linear term), and c (the constant term) from the equation. The term under the square root, \( b^2-4ac \), is known as the discriminant. It's crucial because it determines the number and nature of the roots.

In our exercise, using the quadratic formula involves substituting a = -0.5, b = 12.5, and c = 1.6. The calculation of the roots provides the specific values where the quadratic expression equals zero and helps to delineate intervals on the number line for further analysis. It's important to remember that the roots can be real or complex and that the discriminant's sign indicates this. A positive discriminant implies two real and distinct roots, zero means one real root, and negative one suggests that roots are complex.

The roots of a quadratic equation are the values of x that satisfy the equation; in other words, they make the quadratic expression equal to zero. These roots split the number line into distinct intervals that are examined to find the solution to the inequality. In the given exercise, the two roots obtained after applying the quadratic formula are the critical points of interest.

It's vital to consider the coefficients of the quadratic equation while interpreting these roots. If the coefficient of the x²-term, a, is positive, the parabola opens upwards, and the values between the roots will make the inequality negative (less than zero). Conversely, if a is negative, as in our case, the parabola opens downwards, and values between the roots render the inequality positive (greater than zero). Understanding this fundamental behavior of quadratic functions is essential for determining the solution sets for inequalities.

To determine inequality solution intervals, one must consider the sign of the inequality. For the inequality \( -0.5 x^{2}+12.5 x+1.6>0 \), we are interested in finding where the quadratic expression is greater than zero. After finding the roots of the equation, the number line is divided into intervals delineated by these roots.

To find which intervals satisfy the inequality, select any test point from each interval and substitute it back into the inequality. If the result is positive, the interval is part of the solution set. Conversely, if the result is negative, the interval does not satisfy the inequality. In our exercise, since the gap between the roots leads to a positive outcome, it indicates the part of the number line where the original inequality holds true. Keep in mind the importance of the original inequality: as it does not include an equals sign, the roots themselves are not included in the solution interval.

Plotting a number line is an effective way to visualize the solution intervals of an inequality. It serves as a graphic organizer where the roots of the quadratic equation are marked, typically with open or closed circles to denote whether they are included in the solution set. In this exercise, since the inequality does not include the equal sign, open circles would be used.

On the number line, the areas where the inequality is true will be shaded or marked distinctly. This visual tool helps to clarify the solution set and ensures that one can easily communicate where the original inequality is satisfied. In our example, plotting the roots and shading the appropriate intervals makes it clear which sections of the number line are within the solution set. Students can easily identify the valid intervals at a glance with this method, making the number line a fundamental aspect of solving and representing inequalities.