Solving Quadratic Equations Using The Quadratic Formula Worksheet - Free Printable
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Step-by-step solution for: Solving Quadratic Equations Using The Quadratic Formula Worksheet
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Show Answer Key & Explanations
Step-by-step solution for: Solving Quadratic Equations Using The Quadratic Formula Worksheet
Let’s solve each quadratic equation using the quadratic formula:
> For any equation in the form:
> ax² + bx + c = 0,
> the solutions are:
> x = [-b ± √(b² - 4ac)] / (2a)
We’ll go one by one, carefully.
---
1) 3n² – 5n – 8 = 0
a = 3, b = -5, c = -8
Discriminant: (-5)² - 4(3)(-8) = 25 + 96 = 121 → √121 = 11
n = [5 ± 11] / 6
→ n = (5+11)/6 = 16/6 = 8/3
→ n = (5-11)/6 = -6/6 = -1
✔ Final Answer for #1: n = 8/3 or n = -1
---
2) x² + 10x + 21 = 0
a=1, b=10, c=21
Discriminant: 100 - 84 = 16 → √16 = 4
x = [-10 ± 4]/2
→ x = (-10+4)/2 = -6/2 = -3
→ x = (-10-4)/2 = -14/2 = -7
✔ Final Answer for #2: x = -3 or x = -7
---
3) 10x² – 9x + 6 = 0
a=10, b=-9, c=6
Discriminant: 81 - 240 = -159 → negative → no real solutions
✔ Final Answer for #3: No real solutions
---
4) p² – 9 = 0
This is difference of squares: (p-3)(p+3)=0 → p=3 or p=-3
Or use formula: a=1, b=0, c=-9
Discriminant: 0 + 36 = 36 → √36=6
p = [0 ± 6]/2 → p=3 or p=-3
✔ Final Answer for #4: p = 3 or p = -3
---
5) 6x² – 12x + 1 = 0
a=6, b=-12, c=1
Discriminant: 144 - 24 = 120 → √120 = 2√30
x = [12 ± 2√30]/12 = [6 ± √30]/6
✔ Final Answer for #5: x = (6 + √30)/6 or x = (6 - √30)/6
---
6) 6n² – 11 = 0
a=6, b=0, c=-11
Discriminant: 0 + 264 = 264 → √264 = 2√66
n = [0 ± 2√66]/12 = ±√66/6
✔ Final Answer for #6: n = √66/6 or n = -√66/6
---
7) 2n² + 5n – 9 = 0
a=2, b=5, c=-9
Discriminant: 25 + 72 = 97 → √97 (can’t simplify)
n = [-5 ± √97]/4
✔ Final Answer for #7: n = (-5 + √97)/4 or n = (-5 - √97)/4
---
8) 3x² – 6x – 23 = 0
a=3, b=-6, c=-23
Discriminant: 36 + 276 = 312 → √312 = 2√78
x = [6 ± 2√78]/6 = [3 ± √78]/3
✔ Final Answer for #8: x = (3 + √78)/3 or x = (3 - √78)/3
---
9) 6k² + 12k – 15 = –10
First, move all to left: 6k² + 12k – 5 = 0
a=6, b=12, c=-5
Discriminant: 144 + 120 = 264 → √264 = 2√66
k = [-12 ± 2√66]/12 = [-6 ± √66]/6
✔ Final Answer for #9: k = (-6 + √66)/6 or k = (-6 - √66)/6
---
10) 8x² – 14 = –11
Move all to left: 8x² – 3 = 0
a=8, b=0, c=-3
Discriminant: 0 + 96 = 96 → √96 = 4√6
x = [0 ± 4√6]/16 = ±√6/4
✔ Final Answer for #10: x = √6/4 or x = -√6/4
---
11) 6k² + 2k + 9 = –3
Move all to left: 6k² + 2k + 12 = 0 → divide by 2: 3k² + k + 6 = 0
a=3, b=1, c=6
Discriminant: 1 - 72 = -71 → no real solutions
✔ Final Answer for #11: No real solutions
---
12) 12p² + 9p – 30 = –10
Move all to left: 12p² + 9p – 20 = 0
a=12, b=9, c=-20
Discriminant: 81 + 960 = 1041 → √1041 (doesn’t simplify nicely)
p = [-9 ± √1041]/24
✔ Final Answer for #12: p = (-9 + √1041)/24 or p = (-9 - √1041)/24
---
13) 3x² = –7x + 136
Move all to left: 3x² + 7x – 136 = 0
a=3, b=7, c=-136
Discriminant: 49 + 1632 = 1681 → √1681 = 41
x = [-7 ± 41]/6
→ x = (34)/6 = 17/3
→ x = (-48)/6 = -8
✔ Final Answer for #13: x = 17/3 or x = -8
---
14) 3n² = –n + 14
Move all to left: 3n² + n – 14 = 0
a=3, b=1, c=-14
Discriminant: 1 + 168 = 169 → √169 = 13
n = [-1 ± 13]/6
→ n = 12/6 = 2
→ n = -14/6 = -7/3
✔ Final Answer for #14: n = 2 or n = -7/3
---
15) 6v² + 3 = –2v
Move all to left: 6v² + 2v + 3 = 0
a=6, b=2, c=3
Discriminant: 4 - 72 = -68 → no real solutions
✔ Final Answer for #15: No real solutions
---
16) 9p² – 7 = 9p
Move all to left: 9p² – 9p – 7 = 0
a=9, b=-9, c=-7
Discriminant: 81 + 252 = 333 → √333 = 3√37
p = [9 ± 3√37]/18 = [3 ± √37]/6
✔ Final Answer for #16: p = (3 + √37)/6 or p = (3 - √37)/6
---
17) 11k² + 4k – 52 = 10k² – 7
Move all to left: k² + 4k – 45 = 0
a=1, b=4, c=-45
Discriminant: 16 + 180 = 196 → √196 = 14
k = [-4 ± 14]/2
→ k = 10/2 = 5
→ k = -18/2 = -9
✔ Final Answer for #17: k = 5 or k = -9
---
18) –4a² + 18a – 15 = –7a² + 9a
Move all to left: 3a² + 9a – 15 = 0 → divide by 3: a² + 3a – 5 = 0
a=1, b=3, c=-5
Discriminant: 9 + 20 = 29 → √29
a = [-3 ± √29]/2
✔ Final Answer for #18: a = (-3 + √29)/2 or a = (-3 - √29)/2
---
19) –4n(n – 2) = 6(n + 3) – 11n²
Expand both sides:
Left: –4n² + 8n
Right: 6n + 18 – 11n²
Bring all to left: –4n² + 8n – 6n – 18 + 11n² = 0
→ 7n² + 2n – 18 = 0
a=7, b=2, c=-18
Discriminant: 4 + 504 = 508 → √508 = 2√127
n = [-2 ± 2√127]/14 = [-1 ± √127]/7
✔ Final Answer for #19: n = (-1 + √127)/7 or n = (-1 - √127)/7
---
20) x(x – 3) = –7 – 10x
Expand left: x² – 3x
Right: –7 – 10x
Bring all to left: x² – 3x + 7 + 10x = 0 → x² + 7x + 7 = 0
a=1, b=7, c=7
Discriminant: 49 – 28 = 21 → √21
x = [-7 ± √21]/2
✔ Final Answer for #20: x = (-7 + √21)/2 or x = (-7 - √21)/2
---
Final Answer:
1) n = 8/3 or n = -1
2) x = -3 or x = -7
3) No real solutions
4) p = 3 or p = -3
5) x = (6 + √30)/6 or x = (6 - √30)/6
6) n = √66/6 or n = -√66/6
7) n = (-5 + √97)/4 or n = (-5 - √97)/4
8) x = (3 + √78)/3 or x = (3 - √78)/3
9) k = (-6 + √66)/6 or k = (-6 - √66)/6
10) x = √6/4 or x = -√6/4
11) No real solutions
12) p = (-9 + √1041)/24 or p = (-9 - √1041)/24
13) x = 17/3 or x = -8
14) n = 2 or n = -7/3
15) No real solutions
16) p = (3 + √37)/6 or p = (3 - √37)/6
17) k = 5 or k = -9
18) a = (-3 + √29)/2 or a = (-3 - √29)/2
19) n = (-1 + √127)/7 or n = (-1 - √127)/7
20) x = (-7 + √21)/2 or x = (-7 - √21)/2
> For any equation in the form:
> ax² + bx + c = 0,
> the solutions are:
> x = [-b ± √(b² - 4ac)] / (2a)
We’ll go one by one, carefully.
---
1) 3n² – 5n – 8 = 0
a = 3, b = -5, c = -8
Discriminant: (-5)² - 4(3)(-8) = 25 + 96 = 121 → √121 = 11
n = [5 ± 11] / 6
→ n = (5+11)/6 = 16/6 = 8/3
→ n = (5-11)/6 = -6/6 = -1
✔ Final Answer for #1: n = 8/3 or n = -1
---
2) x² + 10x + 21 = 0
a=1, b=10, c=21
Discriminant: 100 - 84 = 16 → √16 = 4
x = [-10 ± 4]/2
→ x = (-10+4)/2 = -6/2 = -3
→ x = (-10-4)/2 = -14/2 = -7
✔ Final Answer for #2: x = -3 or x = -7
---
3) 10x² – 9x + 6 = 0
a=10, b=-9, c=6
Discriminant: 81 - 240 = -159 → negative → no real solutions
✔ Final Answer for #3: No real solutions
---
4) p² – 9 = 0
This is difference of squares: (p-3)(p+3)=0 → p=3 or p=-3
Or use formula: a=1, b=0, c=-9
Discriminant: 0 + 36 = 36 → √36=6
p = [0 ± 6]/2 → p=3 or p=-3
✔ Final Answer for #4: p = 3 or p = -3
---
5) 6x² – 12x + 1 = 0
a=6, b=-12, c=1
Discriminant: 144 - 24 = 120 → √120 = 2√30
x = [12 ± 2√30]/12 = [6 ± √30]/6
✔ Final Answer for #5: x = (6 + √30)/6 or x = (6 - √30)/6
---
6) 6n² – 11 = 0
a=6, b=0, c=-11
Discriminant: 0 + 264 = 264 → √264 = 2√66
n = [0 ± 2√66]/12 = ±√66/6
✔ Final Answer for #6: n = √66/6 or n = -√66/6
---
7) 2n² + 5n – 9 = 0
a=2, b=5, c=-9
Discriminant: 25 + 72 = 97 → √97 (can’t simplify)
n = [-5 ± √97]/4
✔ Final Answer for #7: n = (-5 + √97)/4 or n = (-5 - √97)/4
---
8) 3x² – 6x – 23 = 0
a=3, b=-6, c=-23
Discriminant: 36 + 276 = 312 → √312 = 2√78
x = [6 ± 2√78]/6 = [3 ± √78]/3
✔ Final Answer for #8: x = (3 + √78)/3 or x = (3 - √78)/3
---
9) 6k² + 12k – 15 = –10
First, move all to left: 6k² + 12k – 5 = 0
a=6, b=12, c=-5
Discriminant: 144 + 120 = 264 → √264 = 2√66
k = [-12 ± 2√66]/12 = [-6 ± √66]/6
✔ Final Answer for #9: k = (-6 + √66)/6 or k = (-6 - √66)/6
---
10) 8x² – 14 = –11
Move all to left: 8x² – 3 = 0
a=8, b=0, c=-3
Discriminant: 0 + 96 = 96 → √96 = 4√6
x = [0 ± 4√6]/16 = ±√6/4
✔ Final Answer for #10: x = √6/4 or x = -√6/4
---
11) 6k² + 2k + 9 = –3
Move all to left: 6k² + 2k + 12 = 0 → divide by 2: 3k² + k + 6 = 0
a=3, b=1, c=6
Discriminant: 1 - 72 = -71 → no real solutions
✔ Final Answer for #11: No real solutions
---
12) 12p² + 9p – 30 = –10
Move all to left: 12p² + 9p – 20 = 0
a=12, b=9, c=-20
Discriminant: 81 + 960 = 1041 → √1041 (doesn’t simplify nicely)
p = [-9 ± √1041]/24
✔ Final Answer for #12: p = (-9 + √1041)/24 or p = (-9 - √1041)/24
---
13) 3x² = –7x + 136
Move all to left: 3x² + 7x – 136 = 0
a=3, b=7, c=-136
Discriminant: 49 + 1632 = 1681 → √1681 = 41
x = [-7 ± 41]/6
→ x = (34)/6 = 17/3
→ x = (-48)/6 = -8
✔ Final Answer for #13: x = 17/3 or x = -8
---
14) 3n² = –n + 14
Move all to left: 3n² + n – 14 = 0
a=3, b=1, c=-14
Discriminant: 1 + 168 = 169 → √169 = 13
n = [-1 ± 13]/6
→ n = 12/6 = 2
→ n = -14/6 = -7/3
✔ Final Answer for #14: n = 2 or n = -7/3
---
15) 6v² + 3 = –2v
Move all to left: 6v² + 2v + 3 = 0
a=6, b=2, c=3
Discriminant: 4 - 72 = -68 → no real solutions
✔ Final Answer for #15: No real solutions
---
16) 9p² – 7 = 9p
Move all to left: 9p² – 9p – 7 = 0
a=9, b=-9, c=-7
Discriminant: 81 + 252 = 333 → √333 = 3√37
p = [9 ± 3√37]/18 = [3 ± √37]/6
✔ Final Answer for #16: p = (3 + √37)/6 or p = (3 - √37)/6
---
17) 11k² + 4k – 52 = 10k² – 7
Move all to left: k² + 4k – 45 = 0
a=1, b=4, c=-45
Discriminant: 16 + 180 = 196 → √196 = 14
k = [-4 ± 14]/2
→ k = 10/2 = 5
→ k = -18/2 = -9
✔ Final Answer for #17: k = 5 or k = -9
---
18) –4a² + 18a – 15 = –7a² + 9a
Move all to left: 3a² + 9a – 15 = 0 → divide by 3: a² + 3a – 5 = 0
a=1, b=3, c=-5
Discriminant: 9 + 20 = 29 → √29
a = [-3 ± √29]/2
✔ Final Answer for #18: a = (-3 + √29)/2 or a = (-3 - √29)/2
---
19) –4n(n – 2) = 6(n + 3) – 11n²
Expand both sides:
Left: –4n² + 8n
Right: 6n + 18 – 11n²
Bring all to left: –4n² + 8n – 6n – 18 + 11n² = 0
→ 7n² + 2n – 18 = 0
a=7, b=2, c=-18
Discriminant: 4 + 504 = 508 → √508 = 2√127
n = [-2 ± 2√127]/14 = [-1 ± √127]/7
✔ Final Answer for #19: n = (-1 + √127)/7 or n = (-1 - √127)/7
---
20) x(x – 3) = –7 – 10x
Expand left: x² – 3x
Right: –7 – 10x
Bring all to left: x² – 3x + 7 + 10x = 0 → x² + 7x + 7 = 0
a=1, b=7, c=7
Discriminant: 49 – 28 = 21 → √21
x = [-7 ± √21]/2
✔ Final Answer for #20: x = (-7 + √21)/2 or x = (-7 - √21)/2
---
Final Answer:
1) n = 8/3 or n = -1
2) x = -3 or x = -7
3) No real solutions
4) p = 3 or p = -3
5) x = (6 + √30)/6 or x = (6 - √30)/6
6) n = √66/6 or n = -√66/6
7) n = (-5 + √97)/4 or n = (-5 - √97)/4
8) x = (3 + √78)/3 or x = (3 - √78)/3
9) k = (-6 + √66)/6 or k = (-6 - √66)/6
10) x = √6/4 or x = -√6/4
11) No real solutions
12) p = (-9 + √1041)/24 or p = (-9 - √1041)/24
13) x = 17/3 or x = -8
14) n = 2 or n = -7/3
15) No real solutions
16) p = (3 + √37)/6 or p = (3 - √37)/6
17) k = 5 or k = -9
18) a = (-3 + √29)/2 or a = (-3 - √29)/2
19) n = (-1 + √127)/7 or n = (-1 - √127)/7
20) x = (-7 + √21)/2 or x = (-7 - √21)/2
Parent Tip: Review the logic above to help your child master the concept of solving quadratic equations by quadratic formula worksheet.
