https://file.aiquickdraw.com/mathbot/file/1726485606571eqqwbh82.jpg

Here are the questions from the image, along with their solutions and detailed workings:

### 1. Solve for \( x \):

#### a) \( x^2 = 4 \)
\[
x = \pm \sqrt{4} = \pm 2
\]
**Solution:** \( x = 2 \) or \( x = -2 \)

#### b) \( \frac{2x + 1}{3} = 10 \)
Multiply both sides by 3:
\[
2x + 1 = 30
\]
Subtract 1 from both sides:
\[
2x = 29
\]
Divide by 2:
\[
x = \frac{29}{2} = 14.5
\]
**Solution:** \( x = 14.5 \)

#### c) \( \frac{3}{x} + 1 = 12x \)
First, subtract 1 from both sides:
\[
\frac{3}{x} = 12x - 1
\]
Multiply both sides by \( x \) to eliminate the fraction:
\[
3 = x(12x - 1)
\]
Expand:
\[
3 = 12x^2 - x
\]
Rearrange to form a quadratic equation:
\[
12x^2 - x - 3 = 0
\]
Solve using the quadratic formula:
\[
x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(12)(-3)}}{2(12)} = \frac{1 \pm \sqrt{1 + 144}}{24} = \frac{1 \pm \sqrt{145}}{24}
\]
**Solution:** \( x = \frac{1 \pm \sqrt{145}}{24} \)

#### d) \( 4x + 2 = 8 \)
Subtract 2 from both sides:
\[
4x = 6
\]
Divide by 4:
\[
x = \frac{6}{4} = 1.5
\]
**Solution:** \( x = 1.5 \)

#### e) \( 2x - 3 = 3b \)
This equation includes an unknown \( b \). Solving for \( x \):
\[
2x = 3b + 3
\]
\[
x = \frac{3b + 3}{2}
\]
**Solution:** \( x = \frac{3b + 3}{2} \) (in terms of \( b \))

#### f) \( x - 4 = 1 \)
Add 4 to both sides:
\[
x = 5
\]
**Solution:** \( x = 5 \)

#### g) \( 3c + 2x = 4b \)
Solve for \( x \):
\[
2x = 4b - 3c
\]
\[
x = \frac{4b - 3c}{2}
\]
**Solution:** \( x = \frac{4b - 3c}{2} \) (in terms of \( b \) and \( c \))

### 2. Solve the following equations:

#### w) \( x^3 = 27 \)
Take the cube root of both sides:
\[
x = \sqrt[3]{27} = 3
\]
**Solution:** \( x = 3 \)

#### i) \( x^2 + x^2 = 16 \)
Combine like terms:
\[
2x^2 = 16
\]
Divide by 2:
\[
x^2 = 8
\]
Take the square root:
\[
x = \pm \sqrt{8} = \pm 2\sqrt{2}
\]
**Solution:** \( x = 2\sqrt{2} \) or \( x = -2\sqrt{2} \)

#### j) \( \frac{x}{2} = 4 \)
Multiply both sides by 2:
\[
x = 8
\]
**Solution:** \( x = 8 \)

#### k) \( \frac{2}{x + 4y} = 1 \)
Multiply both sides by \( x + 4y \):
\[
2 = x + 4y
\]
Solving for \( x \):
\[
x = 2 - 4y
\]
**Solution:** \( x = 2 - 4y \)

Would you like more details on any of these solutions, or do you have further questions? 

Here are 5 related problems to explore:

1. Solve \( x^2 = 9 \).
2. Solve \( \frac{4x + 2}{5} = 3 \).
3. Solve \( \frac{6}{x} + 2 = 5x \).
4. Solve \( 5x - 3 = 7b \).
5. Solve \( 2x^3 + 1 = 9 \).

**Tip:** Always check your solutions by substituting them back into the original equation to ensure they satisfy the equation.

Solve for x in the given set of equations from the image.

Explore step-by-step solutions for various algebraic equations involving linear, quadratic, and cubic terms. This guide helps students in Grades 8-10 solve equations such as x² = 4, 2x + 1 = 10, x³ = 27, and more.

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