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The discriminant is a valuable part of solving quadratic equations. It is represented by the Greek letter \( \Delta \) and is found using the formula \( \Delta = b^2 - 4ac \). This formula uses the coefficients from the standard form of a quadratic equation \( ax^2 + bx + c = 0 \).

The discriminant helps us determine the nature of the roots without actually solving the equation. Here's what different values of the discriminant tell us:

• If \( \Delta > 0 \): There are two distinct real roots. The equation touches the x-axis at two points.
• If \( \Delta = 0 \): There is one real root. The equation touches the x-axis at exactly one point, meaning the parabola is tangent to the axis.
• If \( \Delta < 0 \): There are no real roots, but two complex roots. In this case, the parabola does not intersect the x-axis at all.

In our problem, the discriminant was calculated as 361, which is positive, indicating that there are two distinct real roots for the quadratic equation \( 3t^2 - 11t - 20 = 0 \).

The quadratic formula is a universal method that allows us to find the roots of any quadratic equation. This powerful formula is expressed as:

• \( x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} \)

This formula arises from the process of completing the square, but it conveniently provides a quick path to the solution.

The "\( \pm \)" symbol in the formula indicates that there are often two solutions to a quadratic equation - one from adding the square root of the discriminant and another from subtracting it. This aligns with the fact that quadratic graphs, or parabolas, can intersect the x-axis at two points.

In our exercise, with coefficients \( a = 3 \), \( b = -11 \), and \( \Delta = 361 \), we substitute these values into the quadratic formula to find:

• \( t_1 = \frac{-(-11) + \sqrt{361}}{6} \)
• \( t_2 = \frac{-(-11) - \sqrt{361}}{6} \)

After calculating, these give us the roots \( t_1 = 5 \) and \( t_2 = -\frac{4}{3} \).

The roots of a quadratic equation are the solutions where the equation equals zero. These roots indicate the points at which the parabola intersects the x-axis.

There are generally two roots for a quadratic equation since it's a second-degree polynomial. They can be categorized based on the discriminant:

• Real and distinct roots: These occur when \( \Delta > 0 \). The graph intersects the x-axis at two points.
• Real and equal roots: These occur when \( \Delta = 0 \). The graph just touches the x-axis at one point.
• Complex roots: These happen when \( \Delta < 0 \). There is no real intersection with the x-axis, but complex numbers allow for solutions.

In our specific problem, since the discriminant was 361, we determined the equation \( 3t^2 - 11t - 20 = 0 \) has two real and distinct roots. After applying the quadratic formula, we identified these roots as \( t_1 = 5 \) and \( t_2 = -\frac{4}{3} \). These roots are the solutions that solve the original quadratic equation.