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

I'll solve each radical equation step by step, checking my work carefully.

Problem 1: √(p + 9) = 1
- Square both sides: p + 9 = 1² = 1
- Subtract 9: p = 1 - 9 = -8
- Check: (-8 + 9) = √1 = 1 ✓

Problem 2: 10√(9g) = 60
- Divide both sides by 10: √(9g) = 6
- Square both sides: 9g = 36
- Divide by 9: g = 4
- Check: 10√(9·4) = 10√36 = 10·6 = 60 ✓

Problem 3: √x + √(x + 11) = 11
- Isolate one radical: √(x + 11) = 11 - √x
- Square both sides: x + 11 = 121 - 22√x + x
- Simplify: 11 = 121 - 22√x
- Rearrange: 22√x = 110
- Divide by 22: √x = 5
- Square: x = 25
- Check: √25 + √(25 + 11) = 5 + √36 = 5 + 6 = 11 ✓

Problem 4: ³√(2p + 2) = 3
- Cube both sides: 2p + 2 = 27
- Subtract 2: 2p = 25
- Divide by 2: p = 25/2 = 12.5
- Check: ³√(2·12.5 + 2) = ³√27 = 3 ✓

Problem 5: √(2 - x) + 3 = x + 7
- Isolate radical: √(2 - x) = x + 4
- Square both sides: 2 - x = x² + 8x + 16
- Rearrange: x² + 9x + 14 = 0
- Factor: (x + 7)(x + 2) = 0
- Solutions: x = -7 or x = -2
- Check x = -7: √(2 - (-7)) + 3 = √9 + 3 = 6; -7 + 7 = 0 ✗
- Check x = -2: √(2 - (-2)) + 3 = √4 + 3 = 5; -2 + 7 = 5 ✓
- Answer: x = -2

Problem 6: √(2b + 3) = √(5b - 3)
- Square both sides: 2b + 3 = 5b - 3
- Rearrange: 6 = 3b
- Divide by 3: b = 2
- Check: √(2·2 + 3) = √7 and √(5·2 - 3) = √7 ✓

Problem 7: 2f = √(4f² + 6f - 12)
- Square both sides: 4f² = 4f² + 6f - 12
- Simplify: 0 = 6f - 12
- Solve: f = 2
- Check: 2·2 = 4 and √(4·4 + 6·2 - 12) = √(16 + 12 - 12) = √16 = 4 ✓

Problem 8: √(2t + 15) = t
- Square both sides: 2t + 15 = t²
- Rearrange: t² - 2t - 15 = 0
- Factor: (t - 5)(t + 3) = 0
- Solutions: t = 5 or t = -3
- Check t = 5: √(2·5 + 15) = √25 = 5 ✓
- Check t = -3: (2·(-3) + 15) = √9 = 3 ≠ -3 ✗
- Answer: t = 5

Problem 9: 2√(x + 1) - √(2x) = √(x - 4)
This requires careful algebraic manipulation. Let me isolate and square:
- Move terms: 2√(x + 1) = √(2x) + √(x - 4)
- Square both sides: 4(x + 1) = 2x + 2√(2x(x-4)) + (x - 4)
- Simplify: 4x + 4 = 3x - 4 + 2√(2x² - 8x)
- Rearrange: x + 8 = 2√(2x² - 8x)
- Square again: x² + 16x + 64 = 4(2x² - 8x) = 8x² - 32x
- Rearrange: 7x² - 48x - 64 = 0
- Using quadratic formula: x = (48 ± √(2304 + 1792))/14 = (48 ± √4096)/14 = (48 ± 64)/14
- Solutions: x = 112/14 = 8 or x = -16/14 = -8/7
- Check x = 8: 2√9 - √16 = 6 - 4 = 2; √4 = 2 ✓
- Check x = -8/7: Not valid (negative under radicals)
- Answer: x = 8

Problem 10: 10 + (10m - 1) = 13
- Isolate radical: √(10m - 1) = 3
- Square both sides: 10m - 1 = 9
- Add 1: 10m = 10
- Divide by 10: m = 1
- Check: 10 + (10·1 - 1) = 10 + √9 = 10 + 3 = 13 ✓

Final Answer:
1) p = -8
2) g = 4
3) x = 25
4) p = 12.5 (or 25/2)
5) x = -2
6) b = 2
7) f = 2
8) t = 5
9) x = 8
10) m = 1