https://r2.erweima.ai/mathbot/image/1727023707330e2ehxk1f.jpg

Let's go through each equation one by one and solve for \( x \) as the subject.

### a. \( m = ax^2 \)
- Divide both sides by \( a \): 
  \[
  x^2 = \frac{m}{a}
  \]
- Take the square root:
  \[
  x = \pm \sqrt{\frac{m}{a}}
  \]

### b. \( x^2 - y = m \)
- Add \( y \) to both sides:
  \[
  x^2 = m + y
  \]
- Take the square root:
  \[
  x = \pm \sqrt{m + y}
  \]

### c. \( m = n - x^2 \)
- Subtract \( n \) from both sides:
  \[
  x^2 = n - m
  \]
- Take the square root:
  \[
  x = \pm \sqrt{n - m}
  \]

### d. \( \frac{x^2}{y} = a \)
- Multiply both sides by \( y \):
  \[
  x^2 = ay
  \]
- Take the square root:
  \[
  x = \pm \sqrt{ay}
  \]

### e. \( a = \frac{bx^2}{c} \)
- Multiply both sides by \( c \):
  \[
  ac = bx^2
  \]
- Divide both sides by \( b \):
  \[
  x^2 = \frac{ac}{b}
  \]
- Take the square root:
  \[
  x = \pm \sqrt{\frac{ac}{b}}
  \]

### f. \( a = x^2 - b^2 \)
- Add \( b^2 \) to both sides:
  \[
  x^2 = a + b^2
  \]
- Take the square root:
  \[
  x = \pm \sqrt{a + b^2}
  \]

### g. \( m = \frac{n}{x^2} \)
- Multiply both sides by \( x^2 \):
  \[
  mx^2 = n
  \]
- Divide by \( m \):
  \[
  x^2 = \frac{n}{m}
  \]
- Take the square root:
  \[
  x = \pm \sqrt{\frac{n}{m}}
  \]

### h. \( \sqrt{xy} = m \)
- Square both sides:
  \[
  xy = m^2
  \]
- Divide by \( y \):
  \[
  x = \frac{m^2}{y}
  \]

### i. \( a = \sqrt{5x} \)
- Square both sides:
  \[
  a^2 = 5x
  \]
- Divide by \( 5 \):
  \[
  x = \frac{a^2}{5}
  \]

### j. \( y = \sqrt{x - z} \)
- Square both sides:
  \[
  y^2 = x - z
  \]
- Add \( z \) to both sides:
  \[
  x = y^2 + z
  \]

### k. \( y = \sqrt{x - z} - z \)
- Add \( z \) to both sides:
  \[
  y + z = \sqrt{x - z}
  \]
- Square both sides:
  \[
  (y + z)^2 = x - z
  \]
- Add \( z \) to both sides:
  \[
  x = (y + z)^2 + z
  \]

### l. \( a = b + \frac{c}{\sqrt{x}} \)
- Subtract \( b \) from both sides:
  \[
  a - b = \frac{c}{\sqrt{x}}
  \]
- Multiply both sides by \( \sqrt{x} \):
  \[
  (a - b)\sqrt{x} = c
  \]
- Divide by \( a - b \):
  \[
  \sqrt{x} = \frac{c}{a - b}
  \]
- Square both sides:
  \[
  x = \left( \frac{c}{a - b} \right)^2
  \]

### m. \( a - b\sqrt{x} = m \)
- Subtract \( a \) from both sides:
  \[
  -b\sqrt{x} = m - a
  \]
- Divide by \( -b \):
  \[
  \sqrt{x} = \frac{a - m}{b}
  \]
- Square both sides:
  \[
  x = \left( \frac{a - m}{b} \right)^2
  \]

### n. \( \sqrt{3x - 1} = y \)
- Square both sides:
  \[
  3x - 1 = y^2
  \]
- Add 1 to both sides:
  \[
  3x = y^2 + 1
  \]
- Divide by 3:
  \[
  x = \frac{y^2 + 1}{3}
  \]

### o. \( a = \sqrt{y - 2x} \)
- Square both sides:
  \[
  a^2 = y - 2x
  \]
- Subtract \( y \) from both sides:
  \[
  2x = y - a^2
  \]
- Divide by 2:
  \[
  x = \frac{y - a^2}{2}
  \]

### p. \( y = \frac{a}{\sqrt{4x - b}} \)
- Multiply both sides by \( \sqrt{4x - b} \):
  \[
  y\sqrt{4x - b} = a
  \]
- Divide by \( y \):
  \[
  \sqrt{4x - b} = \frac{a}{y}
  \]
- Square both sides:
  \[
  4x - b = \left( \frac{a}{y} \right)^2
  \]
- Add \( b \) to both sides:
  \[
  4x = \left( \frac{a}{y} \right)^2 + b
  \]
- Divide by 4:
  \[
  x = \frac{\left( \frac{a}{y} \right)^2 + b}{4}
  \]

If you have any questions or would like detailed explanations for any specific part, feel free to ask!

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Here are five related questions:
1. How can we simplify the square root for larger numbers in these equations?
2. What happens when the equation involves cube roots instead of square roots?
3. How can logarithms help in solving equations like \( a = b^x \)?
4. How would solving these equations differ if we used decimals instead of variables?
5. How does factoring play a role in solving quadratic equations?

**Tip:** Always check if your solutions are positive or negative when square roots are involved, as there may be two possible solutions.

Make x the subject of each formula in the provided list of equations.

This set of algebraic problems asks students to make x the subject of various equations involving quadratics, square roots, and basic algebraic manipulation. Detailed step-by-step solutions are provided for each equation, suitable for high school students (Grades 9-12).

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