Solving Radical Equations worksheets - Free Printable

Here are the step-by-step solutions for each equation. Remember, whenever you square both sides to remove a square root, you must check your answer at the end to make sure it isn't an "extraneous solution" (a fake answer that doesn't actually work in the original equation).

1) $\sqrt{m + 4} = \sqrt{2m - 8}$
* Square both sides: $m + 4 = 2m - 8$
* Subtract $m$ from both sides: $4 = m - 8$
* Add 8 to both sides: $12 = m$
* Check: $\sqrt{12+4} = \sqrt{16} = 4$ and $\sqrt{2(12)-8} = \sqrt{16} = 4$. It works.

2) $\sqrt{2m + 3} = \sqrt{12 - m}$
* Square both sides: $2m + 3 = 12 - m$
* Add $m$ to both sides: $3m + 3 = 12$
* Subtract 3 from both sides: $3m = 9$
* Divide by 3: $m = 3$
* Check: $\sqrt{2(3)+3} = \sqrt{9} = 3$ and $\sqrt{12-3} = \sqrt{9} = 3$. It works.

3) $\sqrt{x - 6} = 3$
* Square both sides: $x - 6 = 9$
* Add 6 to both sides: $x = 15$
* Check: $\sqrt{15-6} = \sqrt{9} = 3$. It works.

4) $\sqrt{x} = 5$
* Square both sides: $x = 25$
* Check: $\sqrt{25} = 5$. It works.

5) $\sqrt{5m - 1} = \sqrt{3m + 11}$
* Square both sides: $5m - 1 = 3m + 11$
* Subtract $3m$ from both sides: $2m - 1 = 11$
* Add 1 to both sides: $2m = 12$
* Divide by 2: $m = 6$
* Check: $\sqrt{5(6)-1} = \sqrt{29}$ and $\sqrt{3(6)+11} = \sqrt{29}$. It works.

6) $\sqrt{6 - 5x} = 6$
* Square both sides: $6 - 5x = 36$
* Subtract 6 from both sides: $-5x = 30$
* Divide by -5: $x = -6$
* Check: $\sqrt{6 - 5(-6)} = \sqrt{6 + 30} = \sqrt{36} = 6$. It works.

7) $4\sqrt{16x} = 32$
* Divide both sides by 4: $\sqrt{16x} = 8$
* Square both sides: $16x = 64$
* Divide by 16: $x = 4$
* Check: $4\sqrt{16(4)} = 4\sqrt{64} = 4(8) = 32$. It works.

8) $-6\sqrt{3x - 5} = -12$
* Divide both sides by -6: $\sqrt{3x - 5} = 2$
* Square both sides: $3x - 5 = 4$
* Add 5 to both sides: $3x = 9$
* Divide by 3: $x = 3$
* Check: $-6\sqrt{3(3)-5} = -6\sqrt{4} = -6(2) = -12$. It works.

9) $\sqrt{6m + 2} = \sqrt{-3m + 11}$
* Square both sides: $6m + 2 = -3m + 11$
* Add $3m$ to both sides: $9m + 2 = 11$
* Subtract 2 from both sides: $9m = 9$
* Divide by 9: $m = 1$
* Check: $\sqrt{6(1)+2} = \sqrt{8}$ and $\sqrt{-3(1)+11} = \sqrt{8}$. It works.

10) $\sqrt{6x^2 + 12x + 5} = \sqrt{5}$
* Square both sides: $6x^2 + 12x + 5 = 5$
* Subtract 5 from both sides: $6x^2 + 12x = 0$
* Factor out $6x$: $6x(x + 2) = 0$
* Set each part to 0: $6x = 0$ or $x + 2 = 0$
* Solutions: $x = 0$ and $x = -2$
* Check: Both values result in $\sqrt{5} = \sqrt{5}$. Both work.

11) $\sqrt{7x^2 + 8x + 8} = \sqrt{6x^2 + 10x + 7}$
* Square both sides: $7x^2 + 8x + 8 = 6x^2 + 10x + 7$
* Move everything to the left side: $x^2 - 2x + 1 = 0$
* Factor: $(x - 1)(x - 1) = 0$
* Solution: $x = 1$
* Check: $\sqrt{7+8+8} = \sqrt{23}$ and $\sqrt{6+10+7} = \sqrt{23}$. It works.

12) $\sqrt{3x^2 + 2x - 48} = \sqrt{2x^2}$
* Square both sides: $3x^2 + 2x - 48 = 2x^2$
* Subtract $2x^2$ from both sides: $x^2 + 2x - 48 = 0$
* Factor: Find two numbers that multiply to -48 and add to 2. They are +8 and -6.
* $(x + 8)(x - 6) = 0$
* Solutions: $x = -8$ and $x = 6$
* Check:
* If $x = 6$: $\sqrt{3(36)+12-48} = \sqrt{72}$ and $\sqrt{2(36)} = \sqrt{72}$. (Valid)
* If $x = -8$: $\sqrt{3(64)-16-48} = \sqrt{128}$ and $\sqrt{2(64)} = \sqrt{128}$. (Valid)

13) $\sqrt{6k + 17} = \sqrt{5}$
* Square both sides: $6k + 17 = 5$
* Subtract 17 from both sides: $6k = -12$
* Divide by 6: $k = -2$
* Check: $\sqrt{6(-2)+17} = \sqrt{-12+17} = \sqrt{5}$. It works.

14) $5\sqrt{25x} = 100$
* Divide both sides by 5: $\sqrt{25x} = 20$
* Square both sides: $25x = 400$
* Divide by 25: $x = 16$
* Check: $5\sqrt{25(16)} = 5\sqrt{400} = 5(20) = 100$. It works.

Final Answer:
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, x = -2
11) x = 1
12) x = 6, x = -8
13) k = -2
14) x = 16