Worksheet with ten radical equations to solve for the unknown variable.

Here is the step-by-step solution to each of the 10 radical equations on the worksheet. We will isolate the radical, eliminate it by raising both sides to an appropriate power, solve the resulting equation, and always check for extraneous solutions (since squaring both sides can introduce invalid solutions).

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① √(p + 9) = 1

Step 1: Square both sides
→ p + 9 = 1² = 1

Step 2: Solve for p
→ p = 1 - 9 = -8

Check: √(-8 + 9) = √1 = 1 ✓
✔ Answer: p = -8

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② 10√(9g) = 60

Step 1: Divide both sides by 10
→ √(9g) = 6

Step 2: Square both sides
→ 9g = 36

Step 3: Solve for g
→ g = 36 / 9 = 4

Check: 10√(9×4) = 10√36 = 10×6 = 60 ✓
✔ Answer: g = 4

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③ √x + √(x + 11) = 11

This one has two radicals. Isolate one first.

Step 1: Subtract √x from both sides
→ √(x + 11) = 11 - √x

Step 2: Square both sides
→ x + 11 = (11 - √x)² = 121 - 22√x + x

Step 3: Simplify
→ x + 11 = 121 - 22√x + x
Subtract x from both sides:
→ 11 = 121 - 22√x

Step 4: Solve for √x
→ 22√x = 121 - 11 = 110
→ √x = 5

Step 5: Square both sides
→ x = 25

Check: √25 + √(25 + 11) = 5 + √36 = 5 + 6 = 11 ✓
✔ Answer: x = 25

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④ ∛(2p + 2) = 3

Cube root — cube both sides.

Step 1: Cube both sides
→ 2p + 2 = 3³ = 27

Step 2: Solve for p
→ 2p = 25 → p = 25/2 = 12.5

Check: ∛(2×12.5 + 2) = ∛(25 + 2) = ∛27 = 3 ✓
✔ Answer: p = 25/2 or 12.5

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⑤ √(2 - x) + 3 = x + 7

Step 1: Isolate radical
→ √(2 - x) = x + 7 - 3 = x + 4

Step 2: Square both sides
→ 2 - x = (x + 4)² = x² + 8x + 16

Step 3: Bring all terms to one side
→ 0 = x² + 8x + 16 + x - 2
→ x² + 9x + 14 = 0

Step 4: Factor
→ (x + 7)(x + 2) = 0 → x = -7 or x = -2

Check each:

- x = -7: √(2 - (-7)) + 3 = √9 + 3 = 3 + 3 = 6; RHS: -7 + 7 = 0 → 6 ≠ 0 ✘
- x = -2: √(2 - (-2)) + 3 = √4 + 3 = 2 + 3 = 5; RHS: -2 + 7 = 5 ✓

✔ Answer: x = -2

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⑥ √(2b + 3) = √(5b - 3)

Step 1: Square both sides
→ 2b + 3 = 5b - 3

Step 2: Solve
→ 3 + 3 = 5b - 2b → 6 = 3b → b = 2

Check: √(2×2 + 3) = √7; √(5×2 - 3) = √7 ✓
✔ Answer: b = 2

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⑦ 2f = √(4f² + 6f - 12)

Step 1: Square both sides
→ (2f)² = 4f² + 6f - 12
→ 4f² = 4f² + 6f - 12

Step 2: Subtract 4f² from both sides
→ 0 = 6f - 12 → 6f = 12 → f = 2

Check: LHS: 2×2 = 4; RHS: √(4×4 + 6×2 - 12) = √(16 + 12 - 12) = √16 = 4 ✓
✔ Answer: f = 2

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⑧ √(2t + 15) = t

Step 1: Square both sides
→ 2t + 15 = t²

Step 2: Rearrange
→ t² - 2t - 15 = 0

Step 3: Factor
→ (t - 5)(t + 3) = 0 → t = 5 or t = -3

Check:

- t = 5: √(10 + 15) = √25 = 5 ✓
- t = -3: √(-6 + 15) = √9 = 3 ≠ -3 ✘ (radical gives non-negative result)

✔ Answer: t = 5

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⑨ 2√(x + 1) - √(2x) = √(x - 4)

This is tricky. First, note domain restrictions:

- x + 1 ≥ 0 → x ≥ -1
- 2x ≥ 0 → x ≥ 0
- x - 4 ≥ 0 → x ≥ 4

So x ≥ 4

Now solve:

Step 1: Move one radical to the other side
→ 2√(x + 1) = √(2x) + √(x - 4)

Step 2: Square both sides
Left: [2√(x+1)]² = 4(x+1) = 4x + 4
Right: [√(2x) + √(x - 4)]² = 2x + 2√[2x(x-4)] + (x - 4) = 3x - 4 + 2√[2x(x-4)]

So:
→ 4x + 4 = 3x - 4 + 2√[2x(x-4)]

Step 3: Simplify
→ 4x + 4 - 3x + 4 = 2√[2x(x-4)]
→ x + 8 = 2√[2x(x-4)]

Step 4: Divide both sides by 2
→ (x + 8)/2 = √[2x(x - 4)]

Step 5: Square both sides again
→ [(x + 8)/2]² = 2x(x - 4)
→ (x² + 16x + 64)/4 = 2x² - 8x

Multiply both sides by 4:
→ x² + 16x + 64 = 8x² - 32x

Bring all to one side:
→ 0 = 7x² - 48x - 64

Step 6: Use quadratic formula
a=7, b=-48, c=-64
Discriminant: D = (-48)² - 4×7×(-64) = 2304 + 1792 = 4096 = 64²

x = [48 ± 64]/14

→ x = (48 + 64)/14 = 112/14 = 8
→ x = (48 - 64)/14 = -16/14 = -8/7 (reject — less than 4)

Check x=8:

LHS: 2√(8+1) - √(16) = 2×3 - 4 = 6 - 4 = 2
RHS: √(8 - 4) = √4 = 2 ✓

✔ Answer: x = 8

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⑩ 10 + √(10m - 1) = 13

Step 1: Subtract 10
→ √(10m - 1) = 3

Step 2: Square both sides
→ 10m - 1 = 9

Step 3: Solve
→ 10m = 10 → m = 1

Check: 10 + √(10×1 - 1) = 10 + √9 = 10 + 3 = 13 ✓
✔ Answer: m = 1

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## ✔ Final Answers:

① p = -8
② g = 4
③ x = 25
④ p = 25/2
⑤ x = -2
⑥ b = 2
⑦ f = 2
⑧ t = 5
⑨ x = 8
⑩ m = 1

All solutions checked for validity!