Algebra Practice Problems #2: Solve 10 equations involving quadratic expressions, fractions, and square roots.

Here are the step-by-step solutions for each algebra problem on the worksheet.

1) $2x^2 + 4(x^2 + 3) - 2 = 22$
* Step 1: Distribute the 4 into the parentheses: $4 \cdot x^2 = 4x^2$ and $4 \cdot 3 = 12$.
Equation becomes: $2x^2 + 4x^2 + 12 - 2 = 22$
* Step 2: Combine like terms ($2x^2 + 4x^2 = 6x^2$) and constants ($12 - 2 = 10$).
Equation becomes: $6x^2 + 10 = 22$
* Step 3: Subtract 10 from both sides.
$6x^2 = 12$
* Step 4: Divide by 6.
$x^2 = 2$
* Step 5: Take the square root of both sides.
$x = \sqrt{2}$ or $x = -\sqrt{2}$

2) $3x + 2(x - 4) = 20$
* Step 1: Distribute the 2: $2 \cdot x = 2x$ and $2 \cdot -4 = -8$.
Equation becomes: $3x + 2x - 8 = 20$
* Step 2: Combine $x$ terms: $3x + 2x = 5x$.
Equation becomes: $5x - 8 = 20$
* Step 3: Add 8 to both sides.
$5x = 28$
* Step 4: Divide by 5.
$x = \frac{28}{5}$
* Step 5: Convert to a mixed number. 5 goes into 28 five times with a remainder of 3.
$x = 5 \frac{3}{5}$

3) $13 - 3(x + 2) + 5x = 29$
* Step 1: Distribute the -3 carefully: $-3 \cdot x = -3x$ and $-3 \cdot 2 = -6$.
Equation becomes: $13 - 3x - 6 + 5x = 29$
* Step 2: Combine $x$ terms ($-3x + 5x = 2x$) and constants ($13 - 6 = 7$).
Equation becomes: $2x + 7 = 29$
* Step 3: Subtract 7 from both sides.
$2x = 22$
* Step 4: Divide by 2.
$x = 11$

4) $17 + 2(4 + 2x^2) = 33$
* Step 1: Distribute the 2: $2 \cdot 4 = 8$ and $2 \cdot 2x^2 = 4x^2$.
Equation becomes: $17 + 8 + 4x^2 = 33$
* Step 2: Combine constants: $17 + 8 = 25$.
Equation becomes: $25 + 4x^2 = 33$
* Step 3: Subtract 25 from both sides.
$4x^2 = 8$
* Step 4: Divide by 4.
$x^2 = 2$
* Step 5: Take the square root.
$x = \sqrt{2}$ or $x = -\sqrt{2}$

5) $5x - 12 + 3(x - 1) = -14$
* Step 1: Distribute the 3: $3 \cdot x = 3x$ and $3 \cdot -1 = -3$.
Equation becomes: $5x - 12 + 3x - 3 = -14$
* Step 2: Combine $x$ terms ($5x + 3x = 8x$) and constants ($-12 - 3 = -15$).
Equation becomes: $8x - 15 = -14$
* Step 3: Add 15 to both sides.
$8x = 1$
* Step 4: Divide by 8.
$x = \frac{1}{8}$

6) $3(x^2 + 2) - 5 = 28$
* Step 1: Distribute the 3: $3 \cdot x^2 = 3x^2$ and $3 \cdot 2 = 6$.
Equation becomes: $3x^2 + 6 - 5 = 28$
* Step 2: Combine constants: $6 - 5 = 1$.
Equation becomes: $3x^2 + 1 = 28$
* Step 3: Subtract 1 from both sides.
$3x^2 = 27$
* Step 4: Divide by 3.
$x^2 = 9$
* Step 5: Take the square root. The square root of 9 is 3.
$x = 3$ or $x = -3$

7) $6(x^2 - 4) + 8 - 2x^2 = 7$
* Step 1: Distribute the 6: $6 \cdot x^2 = 6x^2$ and $6 \cdot -4 = -24$.
Equation becomes: $6x^2 - 24 + 8 - 2x^2 = 7$
* Step 2: Combine $x^2$ terms ($6x^2 - 2x^2 = 4x^2$) and constants ($-24 + 8 = -16$).
Equation becomes: $4x^2 - 16 = 7$
* Step 3: Add 16 to both sides.
$4x^2 = 23$
* Step 4: Divide by 4.
$x^2 = \frac{23}{4}$
* Step 5: Take the square root of the fraction. $\sqrt{23}$ cannot be simplified, and $\sqrt{4} = 2$.
$x = \frac{\sqrt{23}}{2}$ or $x = -\frac{\sqrt{23}}{2}$

8) $17 - 2x + 3 + x = 11$
* Step 1: Combine $x$ terms ($-2x + x = -x$) and constants ($17 + 3 = 20$).
Equation becomes: $-x + 20 = 11$
* Step 2: Subtract 20 from both sides.
$-x = -9$
* Step 3: Divide by -1 (or multiply by -1) to make $x$ positive.
$x = 9$

9) $10 + 5x - 4(x - 2) = 31$
* Step 1: Distribute the -4: $-4 \cdot x = -4x$ and $-4 \cdot -2 = +8$.
Equation becomes: $10 + 5x - 4x + 8 = 31$
* Step 2: Combine $x$ terms ($5x - 4x = x$) and constants ($10 + 8 = 18$).
Equation becomes: $x + 18 = 31$
* Step 3: Subtract 18 from both sides.
$x = 13$

10) $4x + 3(x - 3) = 26$
* Step 1: Distribute the 3: $3 \cdot x = 3x$ and $3 \cdot -3 = -9$.
Equation becomes: $4x + 3x - 9 = 26$
* Step 2: Combine $x$ terms: $4x + 3x = 7x$.
Equation becomes: $7x - 9 = 26$
* Step 3: Add 9 to both sides.
$7x = 35$
* Step 4: Divide by 7.
$x = 5$

Final Answer:
1) $x = \pm\sqrt{2}$
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) $x = 5$