Find the discriminant of each quadratic equation using the quadratic formula.

Let's solve the problem step by step.

We are asked to find the value of the discriminant for each quadratic equation. The discriminant of a quadratic equation in the form:

$$
ax^2 + bx + c = 0
$$

is given by the formula:

$$
\Delta = b^2 - 4ac
$$

The discriminant tells us about the nature of the roots:
- If $\Delta > 0$: two distinct real roots
- If $\Delta = 0$: one real root (a repeated root)
- If $\Delta < 0$: no real roots (complex roots)

---

We will compute the discriminant for each equation.

---

1) $-x^2 + x + 1 = 0$

$a = -1$, $b = 1$, $c = 1$

$$
\Delta = (1)^2 - 4(-1)(1) = 1 + 4 = 5
$$

✔ Discriminant: 5

---

2) $2x^2 + x + 5 = 0$

$a = 2$, $b = 1$, $c = 5$

$$
\Delta = (1)^2 - 4(2)(5) = 1 - 40 = -39
$$

✔ Discriminant: -39

---

3) $x^2 + 7x + 4 = 0$

$a = 1$, $b = 7$, $c = 4$

$$
\Delta = (7)^2 - 4(1)(4) = 49 - 16 = 33
$$

✔ Discriminant: 33

---

4) $-x^2 - 3x + 2 = 0$

$a = -1$, $b = -3$, $c = 2$

$$
\Delta = (-3)^2 - 4(-1)(2) = 9 + 8 = 17
$$

✔ Discriminant: 17

---

5) $2x^2 - 4x + 1 = 0$

$a = 2$, $b = -4$, $c = 1$

$$
\Delta = (-4)^2 - 4(2)(1) = 16 - 8 = 8
$$

✔ Discriminant: 8

---

6) $-2x^2 - x + 6 = 0$

$a = -2$, $b = -1$, $c = 6$

$$
\Delta = (-1)^2 - 4(-2)(6) = 1 + 48 = 49
$$

✔ Discriminant: 49

---

7) $x^2 - 11x + 3 = 0$

$a = 1$, $b = -11$, $c = 3$

$$
\Delta = (-11)^2 - 4(1)(3) = 121 - 12 = 109
$$

✔ Discriminant: 109

---

8) $-x^2 - 2x - 9 = 0$

$a = -1$, $b = -2$, $c = -9$

$$
\Delta = (-2)^2 - 4(-1)(-9) = 4 - 36 = -32
$$

✔ Discriminant: -32

---

9) $3x^2 + 6x + 3 = 0$

$a = 3$, $b = 6$, $c = 3$

$$
\Delta = (6)^2 - 4(3)(3) = 36 - 36 = 0
$$

✔ Discriminant: 0

---

10) $13x^2 - 7x + 1 = 0$

$a = 13$, $b = -7$, $c = 1$

$$
\Delta = (-7)^2 - 4(13)(1) = 49 - 52 = -3
$$

✔ Discriminant: -3

---

11) $-x^2 + 6x + 7 = 0$

$a = -1$, $b = 6$, $c = 7$

$$
\Delta = (6)^2 - 4(-1)(7) = 36 + 28 = 64
$$

✔ Discriminant: 64

---

12) $19x^2 + 8x = 0$

Note: This is $19x^2 + 8x + 0 = 0$, so $c = 0$

$a = 19$, $b = 8$, $c = 0$

$$
\Delta = (8)^2 - 4(19)(0) = 64 - 0 = 64
$$

✔ Discriminant: 64

---

13) $3x^2 - 5x - 2 = 0$

$a = 3$, $b = -5$, $c = -2$

$$
\Delta = (-5)^2 - 4(3)(-2) = 25 + 24 = 49
$$

✔ Discriminant: 49

---

14) $-x^2 + 7x - 2 = 0$

$a = -1$, $b = 7$, $c = -2$

$$
\Delta = (7)^2 - 4(-1)(-2) = 49 - 8 = 41
$$

✔ Discriminant: 41

---

15) $-4x^2 + 2x + 1 = 0$

$a = -4$, $b = 2$, $c = 1$

$$
\Delta = (2)^2 - 4(-4)(1) = 4 + 16 = 20
$$

✔ Discriminant: 20

---

✔ Final Answers:

| Problem | Discriminant |
|--------|--------------|
| 1) | 5 |
| 2) | -39 |
| 3) | 33 |
| 4) | 17 |
| 5) | 8 |
| 6) | 49 |
| 7) | 109 |
| 8) | -32 |
| 9) | 0 |
| 10) | -3 |
| 11) | 64 |
| 12) | 64 |
| 13) | 49 |
| 14) | 41 |
| 15) | 20 |

---

🔍 Summary:

- Positive discriminants: equations have two distinct real solutions.
- Zero discriminant: one real solution (repeated).
- Negative discriminant: no real solutions (complex conjugate roots).

Let me know if you'd like the actual solutions (roots) using the quadratic formula too!