Free Printable Discriminant Worksheets for Students - Free Printable

Problem Overview:

The task involves solving two parts related to quadratic equations:

1. Finding the value of the discriminant for each quadratic equation.
2. Using the discriminant to determine the number of real and imaginary solutions for each quadratic equation.

#### Key Concepts:
- A quadratic equation is generally written as:
$$
ax^2 + bx + c = 0
$$
where \(a\), \(b\), and \(c\) are constants, and \(a \neq 0\).

- The discriminant (\(\Delta\)) of a quadratic equation is given by:
$$
\Delta = b^2 - 4ac
$$

- The discriminant determines the nature of the roots:
- If \(\Delta > 0\): The equation has two distinct real solutions.
- If \(\Delta = 0\): The equation has one real solution (a repeated root).
- If \(\Delta < 0\): The equation has two complex (imaginary) solutions.

---

Part 1: Finding the Value of the Discriminant

We will calculate the discriminant (\(\Delta = b^2 - 4ac\)) for each quadratic equation.

#### Example Calculation:
For the equation \(x^2 - x = 0\):
- Rewrite it in standard form: \(x^2 - x + 0 = 0\).
- Here, \(a = 1\), \(b = -1\), and \(c = 0\).
- Calculate the discriminant:
$$
\Delta = b^2 - 4ac = (-1)^2 - 4(1)(0) = 1 - 0 = 1
$$

#### Solutions for All Equations:

1. \(x^2 - x = 0\):
- \(a = 1\), \(b = -1\), \(c = 0\).
- \(\Delta = (-1)^2 - 4(1)(0) = 1\).

2. \(x^2 + 2x - 1 = 0\):
- \(a = 1\), \(b = 2\), \(c = -1\).
- \(\Delta = (2)^2 - 4(1)(-1) = 4 + 4 = 8\).

3. \(x^2 - 3x + 5 = 0\):
- \(a = 1\), \(b = -3\), \(c = 5\).
- \(\Delta = (-3)^2 - 4(1)(5) = 9 - 20 = -11\).

4. \(x^2 - x + 1 = 0\):
- \(a = 1\), \(b = -1\), \(c = 1\).
- \(\Delta = (-1)^2 - 4(1)(1) = 1 - 4 = -3\).

5. \(x^2 - x - 2 = 0\):
- \(a = 1\), \(b = -1\), \(c = -2\).
- \(\Delta = (-1)^2 - 4(1)(-2) = 1 + 8 = 9\).

6. \(x^2 - 4x + 6 = 0\):
- \(a = 1\), \(b = -4\), \(c = 6\).
- \(\Delta = (-4)^2 - 4(1)(6) = 16 - 24 = -8\).

7. \(x^2 + 5x + 2 = 0\):
- \(a = 1\), \(b = 5\), \(c = 2\).
- \(\Delta = (5)^2 - 4(1)(2) = 25 - 8 = 17\).

8. \(2x^2 - 2x - 7 = 0\):
- \(a = 2\), \(b = -2\), \(c = -7\).
- \(\Delta = (-2)^2 - 4(2)(-7) = 4 + 56 = 60\).

9. \(2x^2 + 3x + 9 = 0\):
- \(a = 2\), \(b = 3\), \(c = 9\).
- \(\Delta = (3)^2 - 4(2)(9) = 9 - 72 = -63\).

10. \(2x^2 + 5x - 4 = 0\):
- \(a = 2\), \(b = 5\), \(c = -4\).
- \(\Delta = (5)^2 - 4(2)(-4) = 25 + 32 = 57\).

11. \(5x^2 + x - 2 = 0\):
- \(a = 5\), \(b = 1\), \(c = -2\).
- \(\Delta = (1)^2 - 4(5)(-2) = 1 + 40 = 41\).

12. \(-3x^2 + 6x + 2 = 0\):
- \(a = -3\), \(b = 6\), \(c = 2\).
- \(\Delta = (6)^2 - 4(-3)(2) = 36 + 24 = 60\).

13. \(-4x^2 - 4x + 5 = 0\):
- \(a = -4\), \(b = -4\), \(c = 5\).
- \(\Delta = (-4)^2 - 4(-4)(5) = 16 + 80 = 96\).

14. \(-2x^2 - x + 1 = 0\):
- \(a = -2\), \(b = -1\), \(c = 1\).
- \(\Delta = (-1)^2 - 4(-2)(1) = 1 + 8 = 9\).

15. \(6x^2 - 2x - 3 = 0\):
- \(a = 6\), \(b = -2\), \(c = -3\).
- \(\Delta = (-2)^2 - 4(6)(-3) = 4 + 72 = 76\).

16. \(-5x^2 - 3x + 9 = 0\):
- \(a = -5\), \(b = -3\), \(c = 9\).
- \(\Delta = (-3)^2 - 4(-5)(9) = 9 + 180 = 189\).

17. \(4x^2 + 5x - 4 = 0\):
- \(a = 4\), \(b = 5\), \(c = -4\).
- \(\Delta = (5)^2 - 4(4)(-4) = 25 + 64 = 89\).

18. \(6x^2 - 9x = 0\):
- Rewrite as \(6x^2 - 9x + 0 = 0\).
- \(a = 6\), \(b = -9\), \(c = 0\).
- \(\Delta = (-9)^2 - 4(6)(0) = 81\).

19. \(3x^2 - 5x + 1 = 0\):
- \(a = 3\), \(b = -5\), \(c = 1\).
- \(\Delta = (-5)^2 - 4(3)(1) = 25 - 12 = 13\).

20. \(5x^2 + 6x + 6 = 0\):
- \(a = 5\), \(b = 6\), \(c = 6\).
- \(\Delta = (6)^2 - 4(5)(6) = 36 - 120 = -84\).

21. \(x^2 - 9x - 6x = 0\):
- Simplify to \(x^2 - 15x = 0\).
- Rewrite as \(x^2 - 15x + 0 = 0\).
- \(a = 1\), \(b = -15\), \(c = 0\).
- \(\Delta = (-15)^2 - 4(1)(0) = 225\).

22. \(4x^2 = 8x - 6\):
- Rewrite as \(4x^2 - 8x + 6 = 0\).
- \(a = 4\), \(b = -8\), \(c = 6\).
- \(\Delta = (-8)^2 - 4(4)(6) = 64 - 96 = -32\).

23. \(-4x^2 - 4x + 6 = 0\):
- \(a = -4\), \(b = -4\), \(c = 6\).
- \(\Delta = (-4)^2 - 4(-4)(6) = 16 + 96 = 112\).

24. \(8x^2 - 6x + 3 = 5x^2\):
- Simplify to \(3x^2 - 6x + 3 = 0\).
- \(a = 3\), \(b = -6\), \(c = 3\).
- \(\Delta = (-6)^2 - 4(3)(3) = 36 - 36 = 0\).

25. \(-9x^2 = -8x + 8\):
- Rewrite as \(-9x^2 + 8x - 8 = 0\).
- \(a = -9\), \(b = 8\), \(c = -8\).
- \(\Delta = (8)^2 - 4(-9)(-8) = 64 - 288 = -224\).

26. \(9x^2 + 6x + 6 = 0\):
- \(a = 9\), \(b = 6\), \(c = 6\).
- \(\Delta = (6)^2 - 4(9)(6) = 36 - 216 = -180\).

27. \(9x^2 + 3x - 8 = 0\):
- \(a = 9\), \(b = 3\), \(c = -8\).
- \(\Delta = (3)^2 - 4(9)(-8) = 9 + 288 = 297\).

28. \(-2x^2 = 8x - 14\):
- Rewrite as \(-2x^2 - 8x + 14 = 0\).
- \(a = -2\), \(b = -8\), \(c = 14\).
- \(\Delta = (-8)^2 - 4(-2)(14) = 64 + 112 = 176\).

---

Part 2: Determining the Number of Real and Imaginary Solutions

Using the discriminants calculated above, we determine the nature of the roots for each equation:

1. \(\Delta = 1\): Two real solutions.
2. \(\Delta = 8\): Two real solutions.
3. \(\Delta = -11\): Two imaginary solutions.
4. \(\Delta = -3\): Two imaginary solutions.
5. \(\Delta = 9\): Two real solutions.
6. \(\Delta = -8\): Two imaginary solutions.
7. \(\Delta = 17\): Two real solutions.
8. \(\Delta = 60\): Two real solutions.
9. \(\Delta = -63\): Two imaginary solutions.
10. \(\Delta = 57\): Two real solutions.
11. \(\Delta = 41\): Two real solutions.
12. \(\Delta = 60\): Two real solutions.
13. \(\Delta = 96\): Two real solutions.
14. \(\Delta = 9\): Two real solutions.
15. \(\Delta = 76\): Two real solutions.
16. \(\Delta = 189\): Two real solutions.
17. \(\Delta = 89\): Two real solutions.
18. \(\Delta = 81\): Two real solutions.
19. \(\Delta = 13\): Two real solutions.
20. \(\Delta = -84\): Two imaginary solutions.
21. \(\Delta = 225\): Two real solutions.
22. \(\Delta = -32\): Two imaginary solutions.
23. \(\Delta = 112\): Two real solutions.
24. \(\Delta = 0\): One real solution.
25. \(\Delta = -224\): Two imaginary solutions.
26. \(\Delta = -180\): Two imaginary solutions.
27. \(\Delta = 297\): Two real solutions.
28. \(\Delta = 176\): Two real solutions.

---

Final Answer:

The discriminants and the number of real/imaginary solutions are summarized below:

| Equation Number | Discriminant (\(\Delta\)) | Number of Solutions |
|------------------|---------------------------|---------------------|
| 1 | 1 | 2 real |
| 2 | 8 | 2 real |
| 3 | -11 | 2 imaginary |
| 4 | -3 | 2 imaginary |
| 5 | 9 | 2 real |
| 6 | -8 | 2 imaginary |
| 7 | 17 | 2 real |
| 8 | 60 | 2 real |
| 9 | -63 | 2 imaginary |
| 10 | 57 | 2 real |
| 11 | 41 | 2 real |
| 12 | 60 | 2 real |
| 13 | 96 | 2 real |
| 14 | 9 | 2 real |
| 15 | 76 | 2 real |
| 16 | 189 | 2 real |
| 17 | 89 | 2 real |
| 18 | 81 | 2 real |
| 19 | 13 | 2 real |
| 20 | -84 | 2 imaginary |
| 21 | 225 | 2 real |
| 22 | -32 | 2 imaginary |
| 23 | 112 | 2 real |
| 24 | 0 | 1 real |
| 25 | -224 | 2 imaginary |
| 26 | -180 | 2 imaginary |
| 27 | 297 | 2 real |
| 28 | 176 | 2 real |

$$
\boxed{\text{See detailed calculations above.}}
$$