Solving Radical Equations worksheets - Free Printable

Let's solve each of the radical equations step by step, and check for extraneous solutions, as instructed.

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1) $\sqrt{m + 4} = \sqrt{2m - 8}$

Step 1: Square both sides:
$$
(\sqrt{m + 4})^2 = (\sqrt{2m - 8})^2
\Rightarrow m + 4 = 2m - 8
$$

Step 2: Solve:
$$
m + 4 = 2m - 8 \\
4 + 8 = 2m - m \\
12 = m
$$

Step 3: Check:
- Left: $\sqrt{12 + 4} = \sqrt{16} = 4$
- Right: $\sqrt{2(12) - 8} = \sqrt{24 - 8} = \sqrt{16} = 4$

✔ Valid solution.

Answer: $ \boxed{12} $

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2) $\sqrt{2m + 3} = \sqrt{12 - m}$

Step 1: Square both sides:
$$
2m + 3 = 12 - m
$$

Step 2: Solve:
$$
2m + m = 12 - 3 \\
3m = 9 \Rightarrow m = 3
$$

Step 3: Check:
- Left: $\sqrt{2(3) + 3} = \sqrt{6 + 3} = \sqrt{9} = 3$
- Right: $\sqrt{12 - 3} = \sqrt{9} = 3$

✔ Valid solution.

Answer: $ \boxed{3} $

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3) $\sqrt{x - 6} = 3$

Step 1: Square both sides:
$$
x - 6 = 9 \Rightarrow x = 15
$$

Step 2: Check:
- $\sqrt{15 - 6} = \sqrt{9} = 3$ ✔

Answer: $ \boxed{15} $

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4) $\sqrt{x} = 5$

Step 1: Square both sides:
$$
x = 25
$$

Step 2: Check:
- $\sqrt{25} = 5$ ✔

Answer: $ \boxed{25} $

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5) $\sqrt{5m - 1} = \sqrt{3m + 11}$

Step 1: Square both sides:
$$
5m - 1 = 3m + 11
\Rightarrow 2m = 12 \Rightarrow m = 6
$$

Step 2: Check:
- Left: $\sqrt{5(6) - 1} = \sqrt{30 - 1} = \sqrt{29}$
- Right: $\sqrt{3(6) + 11} = \sqrt{18 + 11} = \sqrt{29}$

✔ Valid.

Answer: $ \boxed{6} $

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6) $\sqrt{6 - 5x} = 6$

Step 1: Square both sides:
$$
6 - 5x = 36 \Rightarrow -5x = 30 \Rightarrow x = -6
$$

Step 2: Check:
- $\sqrt{6 - 5(-6)} = \sqrt{6 + 30} = \sqrt{36} = 6$ ✔

Answer: $ \boxed{-6} $

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7) $4\sqrt{16x} = 32$

Step 1: Divide both sides by 4:
$$
\sqrt{16x} = 8
$$

Step 2: Square both sides:
$$
16x = 64 \Rightarrow x = 4
$$

Step 3: Check:
- $4\sqrt{16(4)} = 4\sqrt{64} = 4 \cdot 8 = 32$ ✔

Answer: $ \boxed{4} $

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8) $-6\sqrt{3x - 5} = -12$

Step 1: Divide both sides by -6:
$$
\sqrt{3x - 5} = 2
$$

Step 2: Square both sides:
$$
3x - 5 = 4 \Rightarrow 3x = 9 \Rightarrow x = 3
$$

Step 3: Check:
- $-6\sqrt{3(3) - 5} = -6\sqrt{9 - 5} = -6\sqrt{4} = -6 \cdot 2 = -12$ ✔

Answer: $ \boxed{3} $

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9) $\sqrt{6m + 2} = \sqrt{-3m + 11}$

Step 1: Square both sides:
$$
6m + 2 = -3m + 11
\Rightarrow 9m = 9 \Rightarrow m = 1
$$

Step 2: Check:
- Left: $\sqrt{6(1) + 2} = \sqrt{8}$
- Right: $\sqrt{-3(1) + 11} = \sqrt{8}$ ✔

Answer: $ \boxed{1} $

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10) $\sqrt{6x^2 + 12x + 5} = \sqrt{5}$

Step 1: Square both sides:
$$
6x^2 + 12x + 5 = 5 \Rightarrow 6x^2 + 12x = 0
\Rightarrow 6x(x + 2) = 0
\Rightarrow x = 0 \text{ or } x = -2
$$

Step 2: Check:

- $x = 0$: $\sqrt{0 + 0 + 5} = \sqrt{5}$ ✔
- $x = -2$: $\sqrt{6(4) + 12(-2) + 5} = \sqrt{24 - 24 + 5} = \sqrt{5}$ ✔

Both valid.

Answer: $ \boxed{0, -2} $

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11) $\sqrt{7x^2 + 8x + 8} = \sqrt{6x^2 + 10x + 7}$

Step 1: Square both sides:
$$
7x^2 + 8x + 8 = 6x^2 + 10x + 7
\Rightarrow x^2 - 2x + 1 = 0
\Rightarrow (x - 1)^2 = 0 \Rightarrow x = 1
$$

Step 2: Check:
- Left: $\sqrt{7(1)^2 + 8(1) + 8} = \sqrt{7 + 8 + 8} = \sqrt{23}$
- Right: $\sqrt{6(1)^2 + 10(1) + 7} = \sqrt{6 + 10 + 7} = \sqrt{23}$ ✔

Answer: $ \boxed{1} $

---

12) $\sqrt{3x^2 + 2x - 48} = \sqrt{2x^2}$

Step 1: Square both sides:
$$
3x^2 + 2x - 48 = 2x^2
\Rightarrow x^2 + 2x - 48 = 0
\Rightarrow (x + 8)(x - 6) = 0
\Rightarrow x = -8 \text{ or } x = 6
$$

Step 2: Check:

- $x = -8$:
Left: $\sqrt{3(64) + 2(-8) - 48} = \sqrt{192 - 16 - 48} = \sqrt{128}$
Right: $\sqrt{2(64)} = \sqrt{128}$ ✔

- $x = 6$:
Left: $\sqrt{3(36) + 12 - 48} = \sqrt{108 + 12 - 48} = \sqrt{72}$
Right: $\sqrt{2(36)} = \sqrt{72}$ ✔

Both valid.

Answer: $ \boxed{-8, 6} $

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13) $\sqrt{6k + 17} = \sqrt{5}$

Step 1: Square both sides:
$$
6k + 17 = 5 \Rightarrow 6k = -12 \Rightarrow k = -2
$$

Step 2: Check:
- $\sqrt{6(-2) + 17} = \sqrt{-12 + 17} = \sqrt{5}$ ✔

Answer: $ \boxed{-2} $

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14) $5\sqrt{25x} = 100$

Step 1: Divide by 5:
$$
\sqrt{25x} = 20
$$

Step 2: Square both sides:
$$
25x = 400 \Rightarrow x = 16
$$

Step 3: Check:
- $5\sqrt{25(16)} = 5\sqrt{400} = 5 \cdot 20 = 100$ ✔

Answer: $ \boxed{16} $

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✔ Final Answers Summary:

| Problem | Answer |
|--------|--------|
| 1 | $12$ |
| 2 | $3$ |
| 3 | $15$ |
| 4 | $25$ |
| 5 | $6$ |
| 6 | $-6$ |
| 7 | $4$ |
| 8 | $3$ |
| 9 | $1$ |
| 10 | $0, -2$ |
| 11 | $1$ |
| 12 | $-8, 6$ |
| 13 | $-2$ |
| 14 | $16$ |

All solutions have been checked for extraneous roots, and none were found. Each solution is valid.

Let me know if you'd like these in a printable format or with more detailed steps!