Let's solve the quadratic equation [tex]\(3x^2 - 5x - 2 = 0\)[/tex] and determine the nature of its solutions.
1. Identify the coefficients: For the given quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex],
- [tex]\(a = 3\)[/tex]
- [tex]\(b = -5\)[/tex]
- [tex]\(c = -2\)[/tex]
2. Calculate the discriminant:
The discriminant ([tex]\(D\)[/tex]) of a quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex] is given by:
[tex]\[
D = b^2 - 4ac
\][/tex]
Plugging in the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
[tex]\[
D = (-5)^2 - 4(3)(-2)
\][/tex]
[tex]\[
D = 25 + 24
\][/tex]
[tex]\[
D = 49
\][/tex]
3. Analyze the discriminant:
The nature of the solutions to the quadratic equation can be determined by the value of the discriminant:
- If [tex]\(D > 0\)[/tex], there are two real and distinct solutions. These solutions will be rational if [tex]\(D\)[/tex] is a perfect square and irrational if [tex]\(D\)[/tex] is not a perfect square.
- If [tex]\(D = 0\)[/tex], there is exactly one real solution.
- If [tex]\(D < 0\)[/tex], there are two complex solutions.
4. Determine the nature of the solutions based on the discriminant value:
In our case, [tex]\(D = 49\)[/tex], which is greater than zero and a perfect square (since [tex]\(49 = 7^2\)[/tex]). Thus, the quadratic equation [tex]\(3x^2 - 5x - 2 = 0\)[/tex] has two real and rational solutions.
Conclusion:
The correct answer is:
- Two rational solutions.
