Free Printable Discriminant Worksheets for Students - Free Printable
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Step-by-step solution for: Free Printable Discriminant Worksheets for Students
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Step-by-step solution for: Free Printable Discriminant Worksheets for Students
You’ve uploaded a math worksheet titled “Quadratic Formula and the Discriminant”. The task is to:
1. Find the value of the discriminant for each quadratic equation.
2. Use the discriminant to state the number of real and imaginary solutions.
---
## ✔ What is the Discriminant?
For any quadratic equation in standard form:
> ax² + bx + c = 0
The discriminant (D) is given by:
> D = b² – 4ac
- D > 0 → Two distinct real solutions
- D = 0 → One real solution (a repeated root)
- D < 0 → Two complex (imaginary) solutions — no real solutions
---
Let’s solve all 28 problems step-by-step.
---
## 🔢 Part 1: Find the Discriminant
We’ll go one by one, identify `a`, `b`, `c`, then compute `D = b² - 4ac`.
---
→ a=1, b=-1, c=0
D = (-1)² - 4(1)(0) = 1
---
→ a=1, b=2, c=-1
D = (2)² - 4(1)(-1) = 4 + 4 = 8
---
→ a=1, b=3, c=5
D = 9 - 20 = -11
---
→ a=1, b=-1, c=4
D = 1 - 16 = -15
---
→ a=1, b=1, c=-2
D = 1 - 4(-2) = 1 + 8 = 9
---
→ a=1, b=4, c=-6
D = 16 - 4(1)(-6) = 16 + 24 = 40
---
→ a=1, b=5, c=2
D = 25 - 8 = 17
---
→ a=2, b=-2, c=-7
D = (-2)² - 4(2)(-7) = 4 + 56 = 60
---
→ a=2, b=3, c=9
D = 9 - 4(2)(9) = 9 - 72 = -63
---
→ a=2, b=5, c=-4
D = 25 - 4(2)(-4) = 25 + 32 = 57
---
→ a=5, b=1, c=-2
D = 1 - 4(5)(-2) = 1 + 40 = 41
---
→ a=-3, b=-6, c=2
D = (-6)² - 4(-3)(2) = 36 + 24 = 60
---
→ a=-4, b=-4, c=5
D = 16 - 4(-4)(5) = 16 + 80 = 96
---
→ a=-2, b=-1, c=-1
D = 1 - 4(-2)(-1) = 1 - 8 = -7
---
→ a=6, b=-2, c=-3
D = 4 - 4(6)(-3) = 4 + 72 = 76
---
→ a=-5, b=-3, c=9
D = 9 - 4(-5)(9) = 9 + 180 = 189
---
→ a=4, b=5, c=-4
D = 25 - 4(4)(-4) = 25 + 64 = 89
---
→ a=8, b=-9, c=0
D = 81 - 0 = 81
---
→ a=3, b=-5, c=1
D = 25 - 12 = 13
---
→ a=5, b=4, c=4
D = 16 - 80 = -64
---
## 🔁 Part 2: Use Discriminant to State Number of Real/Imaginary Solutions
We’ll now use the D values above to classify each equation.
---
> -x² - 2x - 9 = 0 (common typo)
Assuming that:
→ a=-1, b=-2, c=-9
D = (-2)² - 4(-1)(-9) = 4 - 36 = -32 → Two imaginary solutions
---
> -9x² + 8x - 8 = 0
→ a=-9, b=8, c=-8
D = 64 - 4(-9)(-8) = 64 - 288 = -224 → Two imaginary solutions
---
→ a=4, b=-8, c=4
D = 64 - 64 = 0 → One real solution
---
→ a=9, b=6, c=5
D = 36 - 180 = -144 → Two imaginary solutions
---
→ a=-4, b=-4, c=-6
D = 16 - 4(-4)(-6) = 16 - 96 = -80 → Two imaginary solutions
---
→ a=27, b=0, c=-11
D = 0 - 4(27)(-11) = 0 + 1188 = 1188 → Two real solutions
---
> 8x² - 5x² + 6x - 3 = 0 → 3x² + 6x - 3 = 0
→ a=3, b=6, c=-3
D = 36 - 4(3)(-3) = 36 + 36 = 72 → Two real solutions
---
> -2x² -8x -8 = 0 → Divide by -2: x² + 4x + 4 = 0
→ a=1, b=4, c=4
D = 16 - 16 = 0 → One real solution
*(Alternatively, without dividing: a=-2, b=-8, c=-8 → D = 64 - 4(-2)(-8) = 64 - 64 = 0)*
---
## ✔ Final Answers Summary
| Problem | Discriminant |
|---------|--------------|
| 1 | 1 |
| 2 | 8 |
| 3 | -11 |
| 4 | -15 |
| 5 | 9 |
| 6 | 40 |
| 7 | 17 |
| 8 | 60 |
| 9 | -63 |
| 10 | 57 |
| 11 | 41 |
| 12 | 60 |
| 13 | 96 |
| 14 | -7 |
| 15 | 76 |
| 16 | 189 |
| 17 | 89 |
| 18 | 81 |
| 19 | 13 |
| 20 | -64 |
---
| Problem | Discriminant | Type of Solutions |
|---------|--------------|----------------------------------|
| 21 | -32 | Two imaginary solutions |
| 22 | -224 | Two imaginary solutions |
| 23 | 0 | One real solution |
| 24 | -144 | Two imaginary solutions |
| 25 | -80 | Two imaginary solutions |
| 26 | 1188 | Two real solutions |
| 27 | 72 | Two real solutions |
| 28 | 0 | One real solution |
---
## 🎯 Key Takeaway:
The discriminant tells you everything about the nature of the roots without solving the whole equation. It’s a powerful shortcut!
✔ If you’re preparing for a test or quiz — memorize:
> D > 0 → 2 real
> D = 0 → 1 real
> D < 0 → 2 imaginary
---
Let me know if you’d like to see the actual roots using the quadratic formula too! 😊
1. Find the value of the discriminant for each quadratic equation.
2. Use the discriminant to state the number of real and imaginary solutions.
---
## ✔ What is the Discriminant?
For any quadratic equation in standard form:
> ax² + bx + c = 0
The discriminant (D) is given by:
> D = b² – 4ac
📌 Interpretation of the Discriminant:
- D > 0 → Two distinct real solutions
- D = 0 → One real solution (a repeated root)
- D < 0 → Two complex (imaginary) solutions — no real solutions
---
Let’s solve all 28 problems step-by-step.
---
## 🔢 Part 1: Find the Discriminant
We’ll go one by one, identify `a`, `b`, `c`, then compute `D = b² - 4ac`.
---
1) x(x - 1) = 0 → x² - x = 0
→ a=1, b=-1, c=0
D = (-1)² - 4(1)(0) = 1
---
2) x² + 2x - 1 = 0
→ a=1, b=2, c=-1
D = (2)² - 4(1)(-1) = 4 + 4 = 8
---
3) x² + 3x + 5 = 0
→ a=1, b=3, c=5
D = 9 - 20 = -11
---
4) x² - x + 4 = 0
→ a=1, b=-1, c=4
D = 1 - 16 = -15
---
5) x² + x - 2 = 0
→ a=1, b=1, c=-2
D = 1 - 4(-2) = 1 + 8 = 9
---
6) x² + 4x - 6 = 0
→ a=1, b=4, c=-6
D = 16 - 4(1)(-6) = 16 + 24 = 40
---
7) x² + 5x + 2 = 0
→ a=1, b=5, c=2
D = 25 - 8 = 17
---
8) 2x² - 2x - 7 = 0
→ a=2, b=-2, c=-7
D = (-2)² - 4(2)(-7) = 4 + 56 = 60
---
9) 2x² + 3x + 9 = 0
→ a=2, b=3, c=9
D = 9 - 4(2)(9) = 9 - 72 = -63
---
10) 2x² + 5x - 4 = 0
→ a=2, b=5, c=-4
D = 25 - 4(2)(-4) = 25 + 32 = 57
---
11) 5x² + x - 2 = 0
→ a=5, b=1, c=-2
D = 1 - 4(5)(-2) = 1 + 40 = 41
---
12) -3x² - 6x + 2 = 0
→ a=-3, b=-6, c=2
D = (-6)² - 4(-3)(2) = 36 + 24 = 60
---
13) -4x² - 4x + 5 = 0
→ a=-4, b=-4, c=5
D = 16 - 4(-4)(5) = 16 + 80 = 96
---
14) -2x² - x - 1 = 0
→ a=-2, b=-1, c=-1
D = 1 - 4(-2)(-1) = 1 - 8 = -7
---
15) 6x² - 2x - 3 = 0
→ a=6, b=-2, c=-3
D = 4 - 4(6)(-3) = 4 + 72 = 76
---
16) -5x² - 3x + 9 = 0
→ a=-5, b=-3, c=9
D = 9 - 4(-5)(9) = 9 + 180 = 189
---
17) 4x² + 5x - 4 = 0
→ a=4, b=5, c=-4
D = 25 - 4(4)(-4) = 25 + 64 = 89
---
18) 8x² - 9x = 0 → 8x² - 9x + 0 = 0
→ a=8, b=-9, c=0
D = 81 - 0 = 81
---
19) 3x² - 5x + 1 = 0
→ a=3, b=-5, c=1
D = 25 - 12 = 13
---
20) 5x² + 4x + 4 = 0
→ a=5, b=4, c=4
D = 16 - 80 = -64
---
## 🔁 Part 2: Use Discriminant to State Number of Real/Imaginary Solutions
We’ll now use the D values above to classify each equation.
---
21) -x² - 2 - 9 = 0 → Wait! This seems miswritten. Probably meant:
> -x² - 2x - 9 = 0 (common typo)
Assuming that:
→ a=-1, b=-2, c=-9
D = (-2)² - 4(-1)(-9) = 4 - 36 = -32 → Two imaginary solutions
---
22) -9x² = -8x + 8 → Bring all terms to left:
> -9x² + 8x - 8 = 0
→ a=-9, b=8, c=-8
D = 64 - 4(-9)(-8) = 64 - 288 = -224 → Two imaginary solutions
---
23) 4x² = 8x - 4 → 4x² - 8x + 4 = 0
→ a=4, b=-8, c=4
D = 64 - 64 = 0 → One real solution
---
24) 9x² + 6x + 5 = 0
→ a=9, b=6, c=5
D = 36 - 180 = -144 → Two imaginary solutions
---
25) -4x² - 4x = 6 → -4x² -4x -6 = 0
→ a=-4, b=-4, c=-6
D = 16 - 4(-4)(-6) = 16 - 96 = -80 → Two imaginary solutions
---
26) 27x² - 3 - 8 = 0 → 27x² - 11 = 0 → 27x² + 0x -11 = 0
→ a=27, b=0, c=-11
D = 0 - 4(27)(-11) = 0 + 1188 = 1188 → Two real solutions
---
27) 8x² = -6x + 3 + 5x² → Bring all to left:
> 8x² - 5x² + 6x - 3 = 0 → 3x² + 6x - 3 = 0
→ a=3, b=6, c=-3
D = 36 - 4(3)(-3) = 36 + 36 = 72 → Two real solutions
---
28) -2x² - 8x - 14 = -6 → Move -6 to left:
> -2x² -8x -8 = 0 → Divide by -2: x² + 4x + 4 = 0
→ a=1, b=4, c=4
D = 16 - 16 = 0 → One real solution
*(Alternatively, without dividing: a=-2, b=-8, c=-8 → D = 64 - 4(-2)(-8) = 64 - 64 = 0)*
---
## ✔ Final Answers Summary
Part 1: Discriminant Values
| Problem | Discriminant |
|---------|--------------|
| 1 | 1 |
| 2 | 8 |
| 3 | -11 |
| 4 | -15 |
| 5 | 9 |
| 6 | 40 |
| 7 | 17 |
| 8 | 60 |
| 9 | -63 |
| 10 | 57 |
| 11 | 41 |
| 12 | 60 |
| 13 | 96 |
| 14 | -7 |
| 15 | 76 |
| 16 | 189 |
| 17 | 89 |
| 18 | 81 |
| 19 | 13 |
| 20 | -64 |
---
Part 2: Nature of Solutions
| Problem | Discriminant | Type of Solutions |
|---------|--------------|----------------------------------|
| 21 | -32 | Two imaginary solutions |
| 22 | -224 | Two imaginary solutions |
| 23 | 0 | One real solution |
| 24 | -144 | Two imaginary solutions |
| 25 | -80 | Two imaginary solutions |
| 26 | 1188 | Two real solutions |
| 27 | 72 | Two real solutions |
| 28 | 0 | One real solution |
---
## 🎯 Key Takeaway:
The discriminant tells you everything about the nature of the roots without solving the whole equation. It’s a powerful shortcut!
✔ If you’re preparing for a test or quiz — memorize:
> D > 0 → 2 real
> D = 0 → 1 real
> D < 0 → 2 imaginary
---
Let me know if you’d like to see the actual roots using the quadratic formula too! 😊
Parent Tip: Review the logic above to help your child master the concept of the quadratic formula and the discriminant worksheet.
