Question 14.1

Use the discriminant to determine the number and type of solutions of the following quadratic equation.

[tex]\[ -2x^2 + 11x - 15 = 0 \][/tex]

Select the correct answer below:

A. There are two distinct rational solutions.
B. There are two distinct irrational solutions.
C. There are two complex solutions.
D. There is a single rational solution.



Answer :

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.