Using the Discriminant to Solve x^2-2x-48=0

Welcome to understanding the discriminant formula with Spark.E!Every quadratic equation can be written in the standard form: a x squared plus b x plus c equals zero.The discriminant is a special part of the quadratic formula, written as b squared minus four a c.Let's look at a specific example: x squared minus two x minus forty-eight equals zero.In this equation, we need to identify the values of a, b, and c.The coefficient of x squared is 1, so a equals 1. The coefficient of x is negative 2, so b equals negative 2. And the constant term is negative 48, so c equals negative 48.The discriminant helps us determine what kind of solutions, or roots, our equation will have.When the discriminant is positive, the parabola crosses the x-axis at two different points, giving us two real roots.When the discriminant equals zero, the parabola touches the x-axis at exactly one point, giving us a repeated root.And when the discriminant is negative, the parabola doesn't cross the x-axis at all, indicating complex roots.Now that we understand what the discriminant tells us, let's learn how to calculate it.Now let's calculate the discriminant by substituting our values into the formula b squared minus 4ac.First, we substitute negative 2 for b, 1 for a, and negative 48 for c.Let's solve this step by step. First, we calculate negative 2 squared, which is 4.Next, we multiply 4 times 1 times negative 48.This gives us 4 minus negative 192, which is the same as 4 plus 192.Finally, we add 4 and 192 to get our discriminant of 196.Since our discriminant is 196, which is positive, we know this quadratic equation has two real roots.These roots will be found at x equals negative 6 and x equals 8 on our number line.The large value of our discriminant, 196, indicates that these roots are relatively far apart on the number line.For comparison, a quadratic equation with a smaller discriminant, like 3, would have roots that are much closer together.Now that we've calculated our discriminant of 196, we can use this value to find our exact solutions.