Algebra worksheet featuring quadratic equations to solve, numbered 15 through 28.

Let’s solve each equation step by step. These are all quadratic equations that can be solved by isolating the squared term and then taking the square root.

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15) n² – 5 = –4

Add 5 to both sides:
n² = 1
Take square root:
n = ±1

✔ Final Answer for 15: ±1

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16) n² + 8 = 80

Subtract 8 from both sides:
n² = 72
Take square root:
n = ±√72 = ±√(36×2) = ±6√2

But since this is likely expecting simplified radical or decimal? Wait — let’s check if it’s meant to be integer. 72 isn’t a perfect square, but maybe we leave as ±6√2? Actually, looking at other problems, many result in integers. Let me double-check:

Wait — 80 - 8 = 72 → yes. But 72 is not a perfect square. Hmm. Maybe I made a mistake? No, problem says n² + 8 = 80 → n²=72 → n=±√72=±6√2. But perhaps they want exact form? Or maybe I misread?

Wait — let’s look ahead. Problem 17: 7v²+1=29 → 7v²=28 → v²=4 → v=±2. That’s clean.

Problem 18: 10n²+2=292 → 10n²=290 → n²=29 → n=±√29. Not integer.

So some answers will be radicals. But let’s keep going carefully.

Actually, for 16: n² = 72 → simplify √72 = √(36*2) = 6√2 → so answer is ±6√2.

But wait — maybe the problem expects decimal? No, in algebra, we usually leave as simplified radical unless told otherwise.

But let’s check if I copied right: “n² + 8 = 80” → yes.

Alternatively, maybe it’s a typo? But no, we must solve as given.

Hold on — let’s do all problems systematically.

Actually, let me re-solve 16:

n² + 8 = 80
→ n² = 72
→ n = ±√72 = ±6√2

Yes.

But looking at problem 20: 9n² +10 =91 → 9n²=81 → n²=9 → n=±3. Nice.

So some are nice, some aren’t. We’ll proceed.

But wait — perhaps for 16, they expect ±√72? Or simplified? I think simplified radical is best.

But let’s continue and see pattern.

Actually, let me solve all and verify each.

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17) 7v² + 1 = 29

Subtract 1:
7v² = 28
Divide by 7:
v² = 4
v = ±2

✔ Final Answer for 17: ±2

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18) 10n² + 2 = 292

Subtract 2:
10n² = 290
Divide by 10:
n² = 29
n = ±√29

✔ Final Answer for 18: ±√29

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19) 2m² + 10 = 210

Subtract 10:
2m² = 200
Divide by 2:
m² = 100
m = ±10

✔ Final Answer for 19: ±10

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20) 9n² + 10 = 91

Subtract 10:
9n² = 81
Divide by 9:
n² = 9
n = ±3

✔ Final Answer for 20: ±3

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21) 5n² – 7 = 488

Add 7:
5n² = 495
Divide by 5:
n² = 99
n = ±√99 = ±√(9×11) = ±3√11

✔ Final Answer for 21: ±3√11

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22) 8n² – 6 = 306

Add 6:
8n² = 312
Divide by 8:
n² = 39
n = ±√39

✔ Final Answer for 22: ±√39

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23) 10n² – 10 = 470

Add 10:
10n² = 480
Divide by 10:
n² = 48
n = ±√48 = ±√(16×3) = ±4√3

✔ Final Answer for 23: ±4√3

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24) 8n² – 4 = 532

Add 4:
8n² = 536
Divide by 8:
n² = 67
n = ±√67

✔ Final Answer for 24: ±√67

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25) 4r² + 1 = 325

Subtract 1:
4r² = 324
Divide by 4:
r² = 81
r = ±9

✔ Final Answer for 25: ±9

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26) 8b² – 7 = 193

Add 7:
8b² = 200
Divide by 8:
b² = 25
b = ±5

✔ Final Answer for 26: ±5

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27) 2k² – 2 = 144

Add 2:
2k² = 146
Divide by 2:
k² = 73
k = ±√73

✔ Final Answer for 27: ±√73

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28) 3 – 4x² = –85

Subtract 3 from both sides:
–4x² = –88
Divide by –4:
x² = 22
x = ±√22

✔ Final Answer for 28: ±√22

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Now, let’s go back to problem 16: n² + 8 = 80 → n² = 72 → n = ±√72 = ±6√2. Is that acceptable? Yes.

But let me confirm calculation: 6√2 squared is 36 * 2 = 72, plus 8 is 80. Correct.

Similarly, problem 15: n² -5 = -4 → n²=1 → n=±1. Correct.

All others checked out.

So final answers:

15) ±1
16) ±6√2
17) ±2
18) ±√29
19) ±10
20) ±3
21) ±3√11
22) ±√39
23) ±4√3
24) ±√67
25) ±9
26) ±5
27) ±√73
28) ±√22

I think that’s all.

Final Answer:
15) ±1
16) ±6√2
17) ±2
18) ±√29
19) ±10
20) ±3
21) ±3√11
22) ±√39
23) ±4√3
24) ±√67
25) ±9
26) ±5
27) ±√73
28) ±√22