Let’s solve each equation one by one. Remember: when you square both sides to get rid of a square root, you might create an answer that doesn’t actually work in the original equation — those are called “extraneous solutions.” So we always check our answers at the end!
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1) 2 = √(3x + 34)
Step 1: Square both sides to eliminate the square root.
→ (2)² = (√(3x + 34))²
→ 4 = 3x + 34
Step 2: Solve for x.
→ 4 - 34 = 3x
→ -30 = 3x
→ x = -10
Step 3: Check in original equation.
Left side: 2
Right side: √(3*(-10) + 34) = √(-30 + 34) = √4 = 2 →
✔ Works!
✔ Final Answer:
x = -10
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2) 1 + √(k + 1) = 9
Step 1: Subtract 1 from both sides.
→ √(k + 1) = 8
Step 2: Square both sides.
→ k + 1 = 64
Step 3: Solve for k.
→ k = 63
Step 4: Check.
Left side: 1 + √(63 + 1) = 1 + √64 = 1 + 8 = 9 →
✔ Works!
✔ Final Answer:
k = 63
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3) 9√(4x) = 72
Step 1: Divide both sides by 9.
→ √(4x) = 8
Step 2: Square both sides.
→ 4x = 64
Step 3: Solve for x.
→ x = 16
Step 4: Check.
Left side: 9 * √(4*16) = 9 * √64 = 9 * 8 = 72 →
✔ Works!
✔ Final Answer:
x = 16
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4) 11 = 2 + √(80p + 1)
Step 1: Subtract 2 from both sides.
→ 9 = √(80p + 1)
Step 2: Square both sides.
→ 81 = 80p + 1
Step 3: Solve for p.
→ 80 = 80p
→ p = 1
Step 4: Check.
Right side: 2 + √(80*1 + 1) = 2 + √81 = 2 + 9 = 11 →
✔ Works!
✔ Final Answer:
p = 1
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5) 6√(9x) = 18
Step 1: Divide both sides by 6.
→ √(9x) = 3
Step 2: Square both sides.
→ 9x = 9
Step 3: Solve for x.
→ x = 1
Step 4: Check.
Left side: 6 * √(9*1) = 6 * √9 = 6 * 3 = 18 →
✔ Works!
✔ Final Answer:
x = 1
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6) √(-4 - x) = √(3x + 24)
Since both sides are square roots, we can square both sides directly.
Step 1: Square both sides.
→ (-4 - x) = 3x + 24
Step 2: Solve for x.
→ -4 - x = 3x + 24
→ -4 - 24 = 3x + x
→ -28 = 4x
→ x = -7
Step 3: Check in original equation.
Left side: √(-4 - (-7)) = √(-4 + 7) = √3
Right side: √(3*(-7) + 24) = √(-21 + 24) = √3 →
✔ Both equal √3, so it works!
✔ Final Answer:
x = -7
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7) √(2n + 24) = √(-8 - 2n)
Again, both sides are square roots — square both sides.
Step 1: Square both sides.
→ 2n + 24 = -8 - 2n
Step 2: Solve for n.
→ 2n + 2n = -8 - 24
→ 4n = -32
→ n = -8
Step 3: Check in original equation.
Left side: √(2*(-8) + 24) = √(-16 + 24) = √8
Right side: √(-8 - 2*(-8)) = √(-8 + 16) = √8 →
✔ Both equal √8, so it works!
✔ Final Answer:
n = -8
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8) 13 = √(1 - 2n) + 10
Step 1: Subtract 10 from both sides.
→ 3 = √(1 - 2n)
Step 2: Square both sides.
→ 9 = 1 - 2n
Step 3: Solve for n.
→ 9 - 1 = -2n
→ 8 = -2n
→ n = -4
Step 4: Check.
Right side: √(1 - 2*(-4)) + 10 = √(1 + 8) + 10 = √9 + 10 = 3 + 10 = 13 →
✔ Works!
✔ Final Answer:
n = -4
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Final Answer:
1) x = -10
2) k = 63
3) x = 16
4) p = 1
5) x = 1
6) x = -7
7) n = -8
8) n = -4
Parent Tip: Review the logic above to help your child master the concept of solving radical equations worksheet algebra 2.
