To determine the number and type of solutions for the quadratic equation [tex]\(-2x^2 + 11x - 15 = 0\)[/tex], we need to examine the discriminant (Δ) of the equation. The general form of a quadratic equation is given by [tex]\(ax^2 + bx + c = 0\)[/tex], where [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] are coefficients.
For our equation [tex]\(-2x^2 + 11x - 15 = 0\)[/tex]:
- [tex]\(a = -2\)[/tex]
- [tex]\(b = 11\)[/tex]
- [tex]\(c = -15\)[/tex]
The discriminant of a quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex] is calculated using the formula:
[tex]\[
\Delta = b^2 - 4ac
\][/tex]
1. Compute the discriminant:
[tex]\[
\Delta = 11^2 - 4(-2)(-15)
\][/tex]
2. Simplify the expression inside the discriminant:
[tex]\[
11^2 = 121
\][/tex]
[tex]\[
4(-2)(-15) = 4 \times 2 \times 15 = 120
\][/tex]
3. Subtract the products:
[tex]\[
\Delta = 121 - 120 = 1
\][/tex]
4. Analyse the discriminant:
- If [tex]\(\Delta > 0\)[/tex]: There are two distinct real solutions.
- If [tex]\(\Delta = 0\)[/tex]: There is one real solution.
- If [tex]\(\Delta < 0\)[/tex]: There are two complex solutions.
Since [tex]\(\Delta = 1\)[/tex], which is greater than 0, there are two distinct real solutions. Since the discriminant (1) is a perfect square, it indicates that the solutions are rational.
Therefore, the correct answer is:
There are two distinct rational solutions.
