This worksheet provides ten algebraic equations for students to solve, focusing on distribution and combining like terms.

Let's solve each of the algebraic equations step by step. I'll go through each problem, simplify, and solve for the variable $ x $. Remember: if the answer is a fraction, reduce it and convert to a mixed number if needed.

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1.) $ 2x^2 + 4(x + 3) - 2 = 22 $

Step 1: Expand the expression
$ 2x^2 + 4x + 12 - 2 = 22 $

Step 2: Simplify
$ 2x^2 + 4x + 10 = 22 $

Step 3: Subtract 22 from both sides
$ 2x^2 + 4x + 10 - 22 = 0 $
$ 2x^2 + 4x - 12 = 0 $

Step 4: Divide entire equation by 2
$ x^2 + 2x - 6 = 0 $

Step 5: Use quadratic formula:
$ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $, where $ a = 1, b = 2, c = -6 $

$ x = \frac{-2 \pm \sqrt{(2)^2 - 4(1)(-6)}}{2(1)} $
$ x = \frac{-2 \pm \sqrt{4 + 24}}{2} $
$ x = \frac{-2 \pm \sqrt{28}}{2} $
$ x = \frac{-2 \pm 2\sqrt{7}}{2} $
$ x = -1 \pm \sqrt{7} $

✔ Answer: $ x = -1 + \sqrt{7} $ or $ x = -1 - \sqrt{7} $

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2.) $ 3x + 2(x - 4) = 20 $

Step 1: Expand
$ 3x + 2x - 8 = 20 $

Step 2: Combine like terms
$ 5x - 8 = 20 $

Step 3: Add 8 to both sides
$ 5x = 28 $

Step 4: Divide by 5
$ x = \frac{28}{5} = 5\frac{3}{5} $

✔ Answer: $ x = 5\frac{3}{5} $

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3.) $ 13 - 3(x + 2) + 5x = 29 $

Step 1: Expand
$ 13 - 3x - 6 + 5x = 29 $

Step 2: Combine like terms
$ (13 - 6) + (-3x + 5x) = 29 $
$ 7 + 2x = 29 $

Step 3: Subtract 7
$ 2x = 22 $

Step 4: Divide by 2
$ x = 11 $

✔ Answer: $ x = 11 $

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4.) $ 17 + 2(4 + 2x^2) = 33 $

Step 1: Expand
$ 17 + 8 + 4x^2 = 33 $

Step 2: Simplify
$ 25 + 4x^2 = 33 $

Step 3: Subtract 25
$ 4x^2 = 8 $

Step 4: Divide by 4
$ x^2 = 2 $

Step 5: Take square root
$ x = \pm \sqrt{2} $

✔ Answer: $ x = \sqrt{2} $ or $ x = -\sqrt{2} $

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5.) $ 5x - 12 + 3(x - 1) = -14 $

Step 1: Expand
$ 5x - 12 + 3x - 3 = -14 $

Step 2: Combine like terms
$ 8x - 15 = -14 $

Step 3: Add 15
$ 8x = 1 $

Step 4: Divide by 8
$ x = \frac{1}{8} $

✔ Answer: $ x = \frac{1}{8} $

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6.) $ 3(x^2 + 2) - 5 = 28 $

Step 1: Expand
$ 3x^2 + 6 - 5 = 28 $

Step 2: Simplify
$ 3x^2 + 1 = 28 $

Step 3: Subtract 1
$ 3x^2 = 27 $

Step 4: Divide by 3
$ x^2 = 9 $

Step 5: Take square root
$ x = \pm 3 $

✔ Answer: $ x = 3 $ or $ x = -3 $

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7.) $ 6(x^2 - 4) + 8 - 2x^2 = 7 $

Step 1: Expand
$ 6x^2 - 24 + 8 - 2x^2 = 7 $

Step 2: Combine like terms
$ (6x^2 - 2x^2) + (-24 + 8) = 7 $
$ 4x^2 - 16 = 7 $

Step 3: Add 16
$ 4x^2 = 23 $

Step 4: Divide by 4
$ x^2 = \frac{23}{4} $

Step 5: Take square root
$ x = \pm \sqrt{\frac{23}{4}} = \pm \frac{\sqrt{23}}{2} $

✔ Answer: $ x = \frac{\sqrt{23}}{2} $ or $ x = -\frac{\sqrt{23}}{2} $

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8.) $ 17 - 2x + 3 + x = 11 $

Step 1: Combine like terms
$ (17 + 3) + (-2x + x) = 11 $
$ 20 - x = 11 $

Step 2: Subtract 20
$ -x = -9 $

Step 3: Multiply by -1
$ x = 9 $

✔ Answer: $ x = 9 $

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9.) $ 10 + 5x - 4(x - 2) = 31 $

Step 1: Expand
$ 10 + 5x - 4x + 8 = 31 $

Step 2: Combine like terms
$ (10 + 8) + (5x - 4x) = 31 $
$ 18 + x = 31 $

Step 3: Subtract 18
$ x = 13 $

✔ Answer: $ x = 13 $

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10.) $ 4x + 3(x - 3) = 26 $

Step 1: Expand
$ 4x + 3x - 9 = 26 $

Step 2: Combine like terms
$ 7x - 9 = 26 $

Step 3: Add 9
$ 7x = 35 $

Step 4: Divide by 7
$ x = 5 $

✔ Answer: $ x = 5 $

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✔ Final Answers:

| Problem | Answer |
|--------|--------|
| 1.) | $ x = -1 \pm \sqrt{7} $ |
| 2.) | $ x = 5\frac{3}{5} $ |
| 3.) | $ x = 11 $ |
| 4.) | $ x = \pm \sqrt{2} $ |
| 5.) | $ x = \frac{1}{8} $ |
| 6.) | $ x = \pm 3 $ |
| 7.) | $ x = \pm \frac{\sqrt{23}}{2} $ |
| 8.) | $ x = 9 $ |
| 9.) | $ x = 13 $ |
| 10.) | $ x = 5 $ |

Let me know if you'd like these formatted differently or need help understanding any steps!