Solving Radical Equations worksheets - Free Printable

Let's solve each radical equation step by step and check for extraneous solutions (solutions that don't satisfy the original equation due to squaring both sides).

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

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

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

Step 3: Check in original equation:
$$
\sqrt{12 + 4} = \sqrt{2(12) - 8} \\
\sqrt{16} = \sqrt{24 - 8} \\
4 = \sqrt{16} = 4 \quad \text{✓ Valid}
$$

✔ Solution: $m = 12$

---

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:
$$
\sqrt{2(3) + 3} = \sqrt{12 - 3} \\
\sqrt{6 + 3} = \sqrt{9} \\
\sqrt{9} = \sqrt{9} = 3 \quad \text{✓ Valid}
$$

✔ Solution: $m = 3$

---

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

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

Step 2: Solve:
$$
x = 15
$$

Step 3: Check:
$$
\sqrt{15 - 6} = \sqrt{9} = 3 \quad \text{✓ Valid}
$$

✔ Solution: $x = 15$

---

4) $\sqrt{x} = 5$

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

Step 2: Check:
$$
\sqrt{25} = 5 \quad \text{✓ Valid}
$$

✔ Solution: $x = 25$

---

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

Step 1: Square both sides:
$$
5m - 1 = 3m + 11
$$

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

Step 3: Check:
$$
\sqrt{5(6) - 1} = \sqrt{30 - 1} = \sqrt{29} \\
\sqrt{3(6) + 11} = \sqrt{18 + 11} = \sqrt{29} \quad \text{✓ Valid}
$$

✔ Solution: $m = 6$

---

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

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

Step 2: Solve:
$$
-5x = 30 \Rightarrow x = -6
$$

Step 3: Check:
$$
\sqrt{6 - 5(-6)} = \sqrt{6 + 30} = \sqrt{36} = 6 \quad \text{✓ Valid}
$$

✔ Solution: $x = -6$

---

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:
$$
4\sqrt{16(4)} = 4\sqrt{64} = 4 \cdot 8 = 32 \quad \text{✓ Valid}
$$

✔ Solution: $x = 4$

---

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:
$$
-6\sqrt{3(3) - 5} = -6\sqrt{9 - 5} = -6\sqrt{4} = -6 \cdot 2 = -12 \quad \text{✓ Valid}
$$

✔ Solution: $x = 3$

---

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

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

Step 2: Solve:
$$
6m + 3m = 11 - 2 \Rightarrow 9m = 9 \Rightarrow m = 1
$$

Step 3: Check:
$$
\sqrt{6(1) + 2} = \sqrt{8}, \quad \sqrt{-3(1) + 11} = \sqrt{8} \quad \text{✓ Valid}
$$

✔ Solution: $m = 1$

---

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

Step 1: Square both sides:
$$
6x^2 + 12x + 5 = 5
$$

Step 2: Simplify:
$$
6x^2 + 12x = 0 \Rightarrow 6x(x + 2) = 0
\Rightarrow x = 0 \text{ or } x = -2
$$

Step 3: Check both:

- $x = 0$:
$\sqrt{6(0)^2 + 12(0) + 5} = \sqrt{5}$ ✓

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

✔ Solutions: $x = 0$, $x = -2$

---

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

Step 1: Square both sides:
$$
7x^2 + 8x + 8 = 6x^2 + 10x + 7
$$

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

Step 3: Check:
Left: $\sqrt{7(1)^2 + 8(1) + 8} = \sqrt{7 + 8 + 8} = \sqrt{23}$
Right: $\sqrt{6(1)^2 + 10(1) + 7} = \sqrt{6 + 10 + 7} = \sqrt{23}$ ✓

✔ Solution: $x = 1$

---

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

Step 1: Square both sides:
$$
3x^2 + 2x - 48 = 2x^2
$$

Step 2: Simplify:
$$
3x^2 - 2x^2 + 2x - 48 = 0 \Rightarrow x^2 + 2x - 48 = 0
$$

Factor:
$$
(x + 8)(x - 6) = 0 \Rightarrow x = -8 \text{ or } x = 6
$$

Step 3: Check both:

- $x = -8$:
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: $\sqrt{2(36)} = \sqrt{72}$ ✓

✔ Solutions: $x = -8$, $x = 6$

---

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

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

Step 2: Check:
$$
\sqrt{6(-2) + 17} = \sqrt{-12 + 17} = \sqrt{5} \quad \text{✓ Valid}
$$

✔ Solution: $k = -2$

---

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

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

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

Step 3: Check:
$$
5\sqrt{25(16)} = 5\sqrt{400} = 5 \cdot 20 = 100 \quad \text{✓ Valid}
$$

✔ Solution: $x = 16$

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## ✔ Final Answers:

| Problem | Solution |
|--------|---------|
| 1 | $m = 12$ |
| 2 | $m = 3$ |
| 3 | $x = 15$ |
| 4 | $x = 25$ |
| 5 | $m = 6$ |
| 6 | $x = -6$ |
| 7 | $x = 4$ |
| 8 | $x = 3$ |
| 9 | $m = 1$ |
| 10 | $x = 0, -2$ |
| 11 | $x = 1$ |
| 12 | $x = -8, 6$ |
| 13 | $k = -2$ |
| 14 | $x = 16$ |

All solutions have been checked and are valid. No extraneous solutions found.

Let me know if you'd like these written out in a printable format!