Solving Quadratic Equations Using The Quadratic Formula Worksheet - Free Printable

We are given 20 quadratic equations to solve using the quadratic formula:

> For any quadratic equation in standard form:
> ax² + bx + c = 0,
> the solutions are:
> x = [ -b ± √(b² - 4ac) ] / (2a)

We’ll go one by one, putting each equation into standard form (if needed), identifying a, b, c, and applying the formula.

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1) 3n² − 5n − 8 = 0

→ a = 3, b = -5, c = -8
Discriminant D = (-5)² - 4(3)(-8) = 25 + 96 = 121
√D = 11
n = [5 ± 11] / 6
→ n = (5+11)/6 = 16/6 = 8/3
→ n = (5-11)/6 = -6/6 = -1

✔ Solutions: n = 8/3, -1

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2) x² + 10x + 21 = 0

→ a=1, b=10, c=21
D = 100 - 84 = 16 → √D = 4
x = [-10 ± 4]/2
→ x = (-10+4)/2 = -6/2 = -3
→ x = (-10-4)/2 = -14/2 = -7

✔ Solutions: x = -3, -7

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3) 10x² − 9x + 6 = 0

→ a=10, b=-9, c=6
D = 81 - 240 = -159 < 0 → No real solutions

✔ Solutions: No real roots (complex only)

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4) p² − 9 = 0

→ a=1, b=0, c=-9
D = 0 + 36 = 36 → √D = 6
p = [0 ± 6]/2 = ±6/2 = ±3

✔ Solutions: p = 3, -3

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5) 6x² − 12x + 1 = 0

→ a=6, b=-12, c=1
D = 144 - 24 = 120 → √120 = 2√30
x = [12 ± 2√30]/12 = [6 ± √30]/6

✔ Solutions: x = (6 ± √30)/6

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6) 6n² − 11 = 0

→ a=6, b=0, c=-11
D = 0 + 264 = 264 → √264 = 2√66
n = [0 ± 2√66]/12 = ±√66 / 6

✔ Solutions: n = ±√66 / 6

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7) 2n² + 5n − 9 = 0

→ a=2, b=5, c=-9
D = 25 + 72 = 97 → √97 (can’t simplify)
n = [-5 ± √97]/4

✔ Solutions: n = (-5 ± √97)/4

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8) 3x² − 6x − 23 = 0

→ a=3, b=-6, c=-23
D = 36 + 276 = 312 → √312 = 2√78
x = [6 ± 2√78]/6 = [3 ± √78]/3

✔ Solutions: x = (3 ± √78)/3

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9) 6k² + 12k − 15 = −10

First, bring all terms to left:
6k² + 12k − 5 = 0
→ a=6, b=12, c=-5
D = 144 + 120 = 264 → √264 = 2√66
k = [-12 ± 2√66]/12 = [-6 ± √66]/6

✔ Solutions: k = (-6 ± √66)/6

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10) 8x² − 14 = −11

Bring all to left:
8x² − 3 = 0
→ a=8, b=0, c=-3
D = 0 + 96 = 96 → √96 = 4√6
x = [0 ± 4√6]/16 = ±√6 / 4

✔ Solutions: x = ±√6 / 4

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11) 6k² + 2k + 9 = −3

Bring all to left:
6k² + 2k + 12 = 0 → divide by 2: 3k² + k + 6 = 0
→ a=3, b=1, c=6
D = 1 - 72 = -71 < 0 → No real solutions

✔ Solutions: No real roots

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12) 12p² + 9p − 30 = −10

Bring all to left:
12p² + 9p − 20 = 0
→ a=12, b=9, c=-20
D = 81 + 960 = 1041 → √1041 (not a perfect square)
p = [-9 ± √1041]/24

✔ Solutions: p = (-9 ± √1041)/24

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13) 3x² = −7x + 136

Bring all to left:
3x² + 7x − 136 = 0
→ a=3, b=7, c=-136
D = 49 + 1632 = 1681 → √1681 = 41
x = [-7 ± 41]/6
→ x = (34)/6 = 17/3
→ x = (-48)/6 = -8

✔ Solutions: x = 17/3, -8

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14) 3n² = −n + 14

Bring all to left:
3n² + n − 14 = 0
→ a=3, b=1, c=-14
D = 1 + 168 = 169 → √169 = 13
n = [-1 ± 13]/6
→ n = 12/6 = 2
→ n = -14/6 = -7/3

✔ Solutions: n = 2, -7/3

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15) 6v² + 3 = −2v

Bring all to left:
6v² + 2v + 3 = 0
→ a=6, b=2, c=3
D = 4 - 72 = -68 < 0 → No real solutions

✔ Solutions: No real roots

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16) 9p² − 7 = 9p

Bring all to left:
9p² − 9p − 7 = 0
→ a=9, b=-9, c=-7
D = 81 + 252 = 333 → √333 = 3√37
p = [9 ± 3√37]/18 = [3 ± √37]/6

✔ Solutions: p = (3 ± √37)/6

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17) 11k² + 4k − 52 = 10k² − 7

Bring all to left:
11k² − 10k² + 4k − 52 + 7 = 0 → k² + 4k − 45 = 0
→ a=1, b=4, c=-45
D = 16 + 180 = 196 → √196 = 14
k = [-4 ± 14]/2
→ k = 10/2 = 5
→ k = -18/2 = -9

✔ Solutions: k = 5, -9

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18) −4a² + 18a − 15 = −7a² + 9a

Bring all to left:
(-4a² + 7a²) + (18a - 9a) − 15 = 0 → 3a² + 9a − 15 = 0
Divide by 3: a² + 3a − 5 = 0
→ a=1, b=3, c=-5
D = 9 + 20 = 29 → √29
a = [-3 ± √29]/2

✔ Solutions: a = (-3 ± √29)/2

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19) −4n(n − 2) = 6(n + 3) − 11n²

Expand both sides:
Left: -4n² + 8n
Right: 6n + 18 − 11n²
So:
-4n² + 8n = 6n + 18 − 11n²
Bring all to left:
(-4n² + 11n²) + (8n - 6n) − 18 = 0 → 7n² + 2n − 18 = 0
→ a=7, b=2, c=-18
D = 4 + 504 = 508 → √508 = 2√127
n = [-2 ± 2√127]/14 = [-1 ± √127]/7

✔ Solutions: n = (-1 ± √127)/7

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20) x(x − 3) = −7 − 10x

Expand left: x² − 3x
So: x² − 3x = −7 − 10x
Bring all to left:
x² − 3x + 7 + 10x = 0 → x² + 7x + 7 = 0
→ a=1, b=7, c=7
D = 49 - 28 = 21 → √21
x = [-7 ± √21]/2

✔ Solutions: x = (-7 ± √21)/2

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## ✔ Final Answers Summary:

1. n = 8/3, -1
2. x = -3, -7
3. No real solutions
4. p = 3, -3
5. x = (6 ± √30)/6
6. n = ±√66 / 6
7. n = (-5 ± √97)/4
8. x = (3 ± √78)/3
9. k = (-6 ± √66)/6
10. x = ±√6 / 4
11. No real solutions
12. p = (-9 ± √1041)/24
13. x = 17/3, -8
14. n = 2, -7/3
15. No real solutions
16. p = (3 ± √37)/6
17. k = 5, -9
18. a = (-3 ± √29)/2
19. n = (-1 ± √127)/7
20. x = (-7 ± √21)/2

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Let me know if you want these simplified further or expressed as decimals!