Algebra Practice: Multi-Step Equations #2 | Interactive Worksheet ... - Free Printable

Problem: Solve the given algebraic equations. If the answer is a fraction, reduce it and convert it to a mixed number if necessary. Some answers may involve a square root.

#### Equations to Solve:
1. \( 2x + 4(x^2 + 3) - 2 = 22 \)
2. \( 3x + 2(x - 4) = 20 \)
3. \( 13 - 3(x + 2) + 5x = 29 \)
4. \( 17 + 2(4 + 2x^2) = 33 \)
5. \( 5x - 12 + 3(x - 1) = -14 \)
6. \( 3(x^2 + 2) = 28 \)
7. \( 6(x^2 - 4) + 8 - 2x^2 = 7 \)
8. \( 17 - 2x + 3 + x = 11 \)
9. \( 10 + 5x - 4(x - 2) = 31 \)
10. \( 4x + 3(x - 3) = 26 \)

---

Solution:

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

1. Expand and simplify:
\[
2x + 4(x^2 + 3) - 2 = 22
\]
\[
2x + 4x^2 + 12 - 2 = 22
\]
\[
4x^2 + 2x + 10 = 22
\]

2. Move all terms to one side:
\[
4x^2 + 2x + 10 - 22 = 0
\]
\[
4x^2 + 2x - 12 = 0
\]

3. Simplify by dividing through by 2:
\[
2x^2 + x - 6 = 0
\]

4. Factor the quadratic equation:
\[
2x^2 + x - 6 = (2x - 3)(x + 2) = 0
\]

5. Solve for \( x \):
\[
2x - 3 = 0 \quad \text{or} \quad x + 2 = 0
\]
\[
x = \frac{3}{2} \quad \text{or} \quad x = -2
\]

Answer: \( x = \frac{3}{2} \) or \( x = -2 \)

---

#### 2. \( 3x + 2(x - 4) = 20 \)

1. Expand and simplify:
\[
3x + 2(x - 4) = 20
\]
\[
3x + 2x - 8 = 20
\]
\[
5x - 8 = 20
\]

2. Solve for \( x \):
\[
5x = 28
\]
\[
x = \frac{28}{5}
\]

Convert to a mixed number:
\[
x = 5 \frac{3}{5}
\]

Answer: \( x = 5 \frac{3}{5} \)

---

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

1. Expand and simplify:
\[
13 - 3(x + 2) + 5x = 29
\]
\[
13 - 3x - 6 + 5x = 29
\]
\[
13 - 6 + 5x - 3x = 29
\]
\[
7 + 2x = 29
\]

2. Solve for \( x \):
\[
2x = 22
\]
\[
x = 11
\]

Answer: \( x = 11 \)

---

#### 4. \( 17 + 2(4 + 2x^2) = 33 \)

1. Expand and simplify:
\[
17 + 2(4 + 2x^2) = 33
\]
\[
17 + 8 + 4x^2 = 33
\]
\[
25 + 4x^2 = 33
\]

2. Solve for \( x^2 \):
\[
4x^2 = 8
\]
\[
x^2 = 2
\]

3. Solve for \( x \):
\[
x = \pm \sqrt{2}
\]

Answer: \( x = \sqrt{2} \) or \( x = -\sqrt{2} \)

---

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

1. Expand and simplify:
\[
5x - 12 + 3(x - 1) = -14
\]
\[
5x - 12 + 3x - 3 = -14
\]
\[
8x - 15 = -14
\]

2. Solve for \( x \):
\[
8x = 1
\]
\[
x = \frac{1}{8}
\]

Answer: \( x = \frac{1}{8} \)

---

#### 6. \( 3(x^2 + 2) = 28 \)

1. Expand and simplify:
\[
3(x^2 + 2) = 28
\]
\[
3x^2 + 6 = 28
\]

2. Solve for \( x^2 \):
\[
3x^2 = 22
\]
\[
x^2 = \frac{22}{3}
\]

3. Solve for \( x \):
\[
x = \pm \sqrt{\frac{22}{3}}
\]

Answer: \( x = \sqrt{\frac{22}{3}} \) or \( x = -\sqrt{\frac{22}{3}} \)

---

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

1. Expand and simplify:
\[
6(x^2 - 4) + 8 - 2x^2 = 7
\]
\[
6x^2 - 24 + 8 - 2x^2 = 7
\]
\[
4x^2 - 16 = 7
\]

2. Solve for \( x^2 \):
\[
4x^2 = 23
\]
\[
x^2 = \frac{23}{4}
\]

3. Solve for \( x \):
\[
x = \pm \sqrt{\frac{23}{4}}
\]
\[
x = \pm \frac{\sqrt{23}}{2}
\]

Answer: \( x = \frac{\sqrt{23}}{2} \) or \( x = -\frac{\sqrt{23}}{2} \)

---

#### 8. \( 17 - 2x + 3 + x = 11 \)

1. Simplify:
\[
17 - 2x + 3 + x = 11
\]
\[
20 - x = 11
\]

2. Solve for \( x \):
\[
-x = 11 - 20
\]
\[
-x = -9
\]
\[
x = 9
\]

Answer: \( x = 9 \)

---

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

1. Expand and simplify:
\[
10 + 5x - 4(x - 2) = 31
\]
\[
10 + 5x - 4x + 8 = 31
\]
\[
10 + 8 + 5x - 4x = 31
\]
\[
18 + x = 31
\]

2. Solve for \( x \):
\[
x = 13
\]

Answer: \( x = 13 \)

---

#### 10. \( 4x + 3(x - 3) = 26 \)

1. Expand and simplify:
\[
4x + 3(x - 3) = 26
\]
\[
4x + 3x - 9 = 26
\]
\[
7x - 9 = 26
\]

2. Solve for \( x \):
\[
7x = 35
\]
\[
x = 5
\]

Answer: \( x = 5 \)

---

Final Answers:

1. \( x = \frac{3}{2} \) or \( x = -2 \)
2. \( x = 5 \frac{3}{5} \)
3. \( x = 11 \)
4. \( x = \sqrt{2} \) or \( x = -\sqrt{2} \)
5. \( x = \frac{1}{8} \)
6. \( x = \sqrt{\frac{22}{3}} \) or \( x = -\sqrt{\frac{22}{3}} \)
7. \( x = \frac{\sqrt{23}}{2} \) or \( x = -\frac{\sqrt{23}}{2} \)
8. \( x = 9 \)
9. \( x = 13 \)
10. \( x = 5 \)

\[
\boxed{
\begin{array}{ll}
1. & x = \frac{3}{2} \text{ or } x = -2 \\
2. & x = 5 \frac{3}{5} \\
3. & x = 11 \\
4. & x = \sqrt{2} \text{ or } x = -\sqrt{2} \\
5. & x = \frac{1}{8} \\
6. & x = \sqrt{\frac{22}{3}} \text{ or } x = -\sqrt{\frac{22}{3}} \\
7. & x = \frac{\sqrt{23}}{2} \text{ or } x = -\frac{\sqrt{23}}{2} \\
8. & x = 9 \\
9. & x = 13 \\
10. & x = 5 \\
\end{array}
}
\]