Handwritten solutions to radical equations on a math worksheet.

Let’s solve each problem step by step. These are equations with square roots (radicals). The key idea is to isolate the radical and then square both sides to get rid of it. After solving, we must check our answer in the original equation — sometimes squaring can give us “fake” answers that don’t actually work!

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Problem 1:

√(7x + 8) = 35

Step 1: Square both sides to remove the square root.
→ (√(7x + 8))² = 35²
→ 7x + 8 = 1225

Step 2: Subtract 8 from both sides.
→ 7x = 1225 - 8 = 1217

Step 3: Divide by 7.
→ x = 1217 ÷ 7 = 173.857...? Wait — let me recalculate.

Wait! 35 squared is 1225? Let me check:
35 × 35 = (30+5)(30+5) = 900 + 300 + 25 = 1225 → correct.

But 1225 - 8 = 1217 → yes.

Now 1217 ÷ 7:
7 × 173 = 1211 → 1217 - 1211 = 6 → so x = 173 and 6/7? That doesn’t match the handwritten answer of x=35.

Wait — maybe I misread the problem? Let me look again.

Actually, looking at the image, the first problem says:

> √(7x + 8) = 35
> Answer: x = 35

But if x = 35, plug into left side:
√(7×35 + 8) = √(245 + 8) = √253 ≈ 15.9 — not 35. So something’s wrong.

Wait — perhaps the problem was meant to be √(7x + 8) = √something else? Or maybe it's a typo?

Hold on — let me re-express. Maybe the student wrote the answer incorrectly? Or maybe the problem is different?

Looking back at the image — actually, in the image, for problem 1, they have:

> √(7x + 8) = 35
> Then they write: x = 35

That can't be right. Unless... wait — maybe the equation is √(7x + 8) = √(something)? No.

Alternatively — perhaps the problem is √(7x + 8) = 35, but they solved it as:

They might have done:
7x + 8 = 35 → then 7x = 27 → x = 27/7 — no.

Wait — let me try solving correctly:

√(7x + 8) = 35
Square both sides:
7x + 8 = 1225
7x = 1217
x = 1217 / 7 = 173.857...

But that’s not an integer. And the student wrote x=35. That suggests maybe the problem was supposed to be:

√(7x + 8) = √(253) or something? Not helpful.

Wait — perhaps the problem is √(7x + 8) = 35, but the student made a mistake? But the instruction says "solve each equation" — so we should solve correctly.

But let’s check problem 2:

Problem 2: √(3x + 20) = 31

Student answer: x = 27

Check: √(3*27 + 20) = √(81 + 20) = √101 ≈ 10.05 — not 31. Wrong.

Wait — this is confusing. Let me look at problem 3:

Problem 3: √(x + 7) = 30

Student answer: x = 29

Check: √(29 + 7) = √36 = 6 — not 30. Way off.

This suggests that the students’ answers are incorrect, or perhaps the problems are written differently.

Wait — looking more carefully at the image — in problem 1, it says:

> √(7x + 8) = 35
> Then below, they have:
> x = 35

But that’s impossible. Unless... oh! Wait a minute — perhaps the equation is not √(7x + 8) = 35, but rather √(7x + 8) = √(35^2) or something? No.

Another possibility: maybe the number on the right is under the square root too? Like √(7x + 8) = √35? But that would be unusual.

Or perhaps it's a formatting issue — maybe it's √(7x) + 8 = 35? Let’s try that.

If √(7x) + 8 = 35
Then √(7x) = 27
Square both sides: 7x = 729
x = 729 / 7 ≈ 104.14 — still not 35.

What if it's √(7(x + 8)) = 35?
Then 7(x+8) = 1225
x+8 = 175
x = 167 — not 35.

I think there might be a misunderstanding. Let me look at problem 4:

Problem 4: √(x - 2) = 8
Student answer: x = 10

Check: √(10 - 2) = √8 ≈ 2.828 — not 8. Should be x = 66, because √(66-2)=√64=8.

Ah! Here’s the pattern — the student is writing the answer as the number inside the square root after squaring, but forgetting to add/subtract properly.

For example, in problem 4:
√(x - 2) = 8
Square both sides: x - 2 = 64
Then x = 66 — but student wrote x=10, which is 8+2? No, 8+2=10, but that’s wrong because you need to square first.

Similarly, problem 5: √(2x + 1) = 15
Student answer: x = 10

Correct solution:
2x + 1 = 225
2x = 224
x = 112 — not 10.

Student probably did: 2x + 1 = 15 → 2x=14 → x=7, but wrote 10? Inconsistent.

Problem 6: √(x + 9) = 6
Student answer: x = 60

Correct: x + 9 = 36 → x = 27 — not 60.

This is messy. Perhaps the problems are miswritten in the image, or the students made consistent errors.

But let’s take problem 7:
√(x + 1) = 9
Student answer: x = 80

Correct: x + 1 = 81 → x = 80 — YES! This one is correct.

Problem 8: √(2x - 4) = 12
Student answer: x = 11

Correct: 2x - 4 = 144 → 2x = 148 → x = 74 — not 11. Student probably did 2x - 4 = 12 → 2x=16 → x=8, but wrote 11? No.

Problem 9: √(7x + 7) = 7
Student answer: x = 60

Correct: 7x + 7 = 49 → 7x = 42 → x = 6 — not 60.

Problem 10: √(3x + 7) = 49
Student answer: x = 9

Correct: 3x + 7 = 2401 → 3x = 2394 → x = 798 — not 9.

Problem 11: √(9x + 1) = 28
Student answer: x = 9

Correct: 9x + 1 = 784 → 9x = 783 → x = 87 — not 9.

Problem 12: √(12x + 10) = 46
Student answer: x = 6

Correct: 12x + 10 = 2116 → 12x = 2106 → x = 175.5 — not 6.

Only problem 7 is correct among the student answers.

But the task is to "solve each equation", so I should provide the correct solutions, not the student's answers.

Let me solve them correctly one by one.

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Correct Solutions:

#### 1. √(7x + 8) = 35
Square both sides:
7x + 8 = 1225
7x = 1217
x = 1217 / 7 = 173.857... — but let's keep as fraction: 1217/7

But 1217 ÷ 7 = 173 with remainder 6, so x = 173 6/7

However, let's verify:
√(7*(1217/7) + 8) = √(1217 + 8) = √1225 = 35 — correct.

So x = 1217/7 or approximately 173.857

But since other problems have integer answers, perhaps there's a typo in the problem. Maybe it's √(7x + 8) = √1225, but that's the same.

Perhaps the problem is √(7x) + 8 = 35? Let's assume that for now, but better to stick to what's written.

Given the context, and that only problem 7 matches, I think we should solve as written.

But to make it clean, let's do all correctly.

#### 2. √(3x + 20) = 31
Square: 3x + 20 = 961
3x = 941
x = 941/3 ≈ 313.666...

Verify: √(3*941/3 + 20) = √(941 + 20) = √961 = 31 — correct.

#### 3. √(x + 7) = 30
Square: x + 7 = 900
x = 893

Verify: √(893 + 7) = √900 = 30 — correct.

#### 4. √(x - 2) = 8
Square: x - 2 = 64
x = 66

Verify: √(66 - 2) = √64 = 8 — correct.

#### 5. √(2x + 1) = 15
Square: 2x + 1 = 225
2x = 224
x = 112

Verify: √(2*112 + 1) = √(224 + 1) = √225 = 15 — correct.

#### 6. √(x + 9) = 6
Square: x + 9 = 36
x = 27

Verify: √(27 + 9) = √36 = 6 — correct.

#### 7. √(x + 1) = 9
Square: x + 1 = 81
x = 80

Verify: √(80 + 1) = √81 = 9 — correct. (Matches student answer)

#### 8. √(2x - 4) = 12
Square: 2x - 4 = 144
2x = 148
x = 74

Verify: √(2*74 - 4) = √(148 - 4) = √144 = 12 — correct.

#### 9. √(7x + 7) = 7
Square: 7x + 7 = 49
7x = 42
x = 6

Verify: √(7*6 + 7) = √(42 + 7) = √49 = 7 — correct.

#### 10. √(3x + 7) = 49
Square: 3x + 7 = 2401
3x = 2394
x = 798

Verify: √(3*798 + 7) = √(2394 + 7) = √2401 = 49 — correct.

#### 11. √(9x + 1) = 28
Square: 9x + 1 = 784
9x = 783
x = 87

Verify: √(9*87 + 1) = √(783 + 1) = √784 = 28 — correct.

#### 12. √(12x + 10) = 46
Square: 12x + 10 = 2116
12x = 2106
x = 2106 / 12 = 175.5 or 351/2

Verify: √(12*175.5 + 10) = √(2106 + 10) = √2116 = 46 — correct.

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Now, comparing to student answers:

- Only problem 7 is correct (x=80)
- Others are wrong; students likely forgot to square the right-hand side or made arithmetic errors.

Since the task is to solve the equations, I'll provide the correct answers.

Final Answer:
1. x = 1217/7 or approximately 173.857
2. x = 941/3 or approximately 313.667
3. x = 893
4. x = 66
5. x = 112
6. x = 27
7. x = 80
8. x = 74
9. x = 6
10. x = 798
11. x = 87
12. x = 351/2 or 175.5

But since the problems might expect exact fractions or integers, and some are not integers, but in the context, perhaps we leave as simplified fractions.

However, looking back, in problem 1, if it were √(7x + 8) = 35, and 35^2=1225, 1225-8=1217, 1217÷7=173.857..., which is fine.

But to match the format, and since the student answers are mostly integers, perhaps there's a different interpretation.

Another thought: in some problems, the right-hand side might be under the square root? For example, problem 1: √(7x + 8) = √35? But that would be unusual.

Or perhaps the number is the result after squaring? No.

I think we have to go with the mathematical solution.

So for the final answer, I'll list the correct values.

But the user might expect the answers as per the worksheet, but the instruction is to solve accurately.

Perhaps the problems are:

Let me double-check problem 1 in the image — it says "√(7x + 8) = 35" and student wrote x=35, which is wrong.

Similarly, problem 2: "√(3x + 20) = 31" student x=27 — wrong.

But in problem 7, it's correct.

So for accuracy, I'll provide the correct solutions.

Final Answer:
1. \frac{1217}{7}
2. \frac{941}{3}
3. 893
4. 66
5. 112
6. 27
7. 80
8. 74
9. 6
10. 798
11. 87
12. \frac{351}{2}

To make it neat, and since the student might have intended integer answers, but based on calculation, these are correct.

Perhaps in problem 1, it's √(7x) + 8 = 35, then √(7x) = 27, 7x = 729, x = 104.142... still not integer.

Or √(7(x+8)) = 35, then 7(x+8) = 1225, x+8 = 175, x=167.

None give 35.

I think we must conclude that the student answers are incorrect except for #7, and provide the correct ones.

So for the response, I'll box the correct answers.

But the format requires only the final answer after reasoning.

So:

Final Answer:
1. \frac{1217}{7}
2. \frac{941}{3}
3. 893
4. 66
5. 112
6. 27
7. 80
8. 74
9. 6
10. 798
11. 87
12. \frac{351}{2}

To write it cleanly:

Since the problems are numbered, and to match the expected format, I'll list them as such.

Note: In some cases, the answer can be left as a fraction or decimal, but fraction is exact.

For problem 12, 2106 ÷ 12 = 1053 ÷ 6 = 351 ÷ 2, so \frac{351}{2}.

Yes.

So the final answer is the list above.