Match each value of the discriminant with the correct description of the number and type of solutions. Prompts Answers (1) Discriminant \( =25 \) (2) Discriminant \( =30 \) (3) Discriminant \( =0 \) two non-real (imaginary) solutions match (4) Discriminant \( =-9 \) \( \begin{array}{l}\text { one rational solution } \\ \text { two ratrational solutions solutions }\end{array} \)

To match each value of the discriminant with the correct description of the number and type of solutions, we need to understand how the discriminant (\(D\)) of a quadratic equation \(ax^2 + bx + c = 0\) determines the nature of the roots: 1. **If \(D > 0\)**: There are two distinct real solutions. 2. **If \(D = 0\)**: There is exactly one real solution (a repeated root). 3. **If \(D < 0\)**: There are two non-real (imaginary) solutions. Now, let's analyze each discriminant value provided: 1. **Discriminant \( = 25 \)**: Since \(25 > 0\), this corresponds to **two distinct real solutions**. 2. **Discriminant \( = 30 \)**: Since \(30 > 0\), this also corresponds to **two distinct real solutions**. 3. **Discriminant \( = 0 \)**: This corresponds to **one real solution** (a repeated root). 4. **Discriminant \( = -9 \)**: Since \(-9 < 0\), this corresponds to **two non-real (imaginary) solutions**. Now, let's match the discriminants with the descriptions: - (1) Discriminant \( = 25 \) → **two distinct real solutions** - (2) Discriminant \( = 30 \) → **two distinct real solutions** - (3) Discriminant \( = 0 \) → **one rational solution** - (4) Discriminant \( = -9 \) → **two non-real (imaginary) solutions** Thus, the matches are: - (1) Discriminant \( = 25 \) → two distinct real solutions - (2) Discriminant \( = 30 \) → two distinct real solutions - (3) Discriminant \( = 0 \) → one rational solution - (4) Discriminant \( = -9 \) → two non-real (imaginary) solutions