To solve the given quadratic equation [tex]\(3t^2 - 5t = 0\)[/tex], we need to find the discriminant and determine the nature of the solutions. Here's how we can do that step-by-step:
1. Identify the coefficients:
The given quadratic equation is in the form [tex]\(at^2 + bt + c = 0\)[/tex]. Here:
- [tex]\(a = 3\)[/tex]
- [tex]\(b = -5\)[/tex]
- [tex]\(c = 0\)[/tex]
2. Calculate the discriminant:
The discriminant ([tex]\(\Delta\)[/tex]) of a quadratic equation [tex]\(at^2 + bt + c = 0\)[/tex] is given by the formula:
[tex]\[
\Delta = b^2 - 4ac
\][/tex]
In our case, we substitute [tex]\(a = 3\)[/tex], [tex]\(b = -5\)[/tex], and [tex]\(c = 0\)[/tex]:
[tex]\[
\Delta = (-5)^2 - 4 \cdot 3 \cdot 0
\][/tex]
[tex]\[
\Delta = 25 - 0
\][/tex]
[tex]\[
\Delta = 25
\][/tex]
3. Determine the nature of the solutions:
The nature of the solutions depends on the value of the discriminant:
- If [tex]\(\Delta > 0\)[/tex], there are two different real-number solutions.
- If [tex]\(\Delta = 0\)[/tex], there is one real-number solution.
- If [tex]\(\Delta < 0\)[/tex], there are two different imaginary number solutions.
Since [tex]\(\Delta = 25\)[/tex] and [tex]\(25 > 0\)[/tex], the quadratic equation [tex]\(3t^2 - 5t = 0\)[/tex] has two different real-number solutions.
So, the discriminant of [tex]\(3t^2 - 5t = 0\)[/tex] is [tex]\(25\)[/tex], and the equation has two different real-number solutions.
