4x ^ 2 - 9 = 0 - 0.1x ^ 2 + 10 = 0 6v ^ 2 + 24 = 0 - x ^ 2 + 3 = 0 y ^ 2 - 1/9 = 0 3m ^ 2 - 1 = 0

Let's solve each equation step-by-step.

1. Equation: \(4x^2 - 9 = 0\)

   Rearranging gives: \[ 4x^2 = 9 \] Dividing by 4: \[ x^2 = \frac{9}{4} \] Taking the square root: \[ x = \pm \frac{3}{2} \]

2. Equation: \(-0.1x^2 + 10 = 0\)

   Rearranging gives: \[ 0.1x^2 = 10 \] Dividing by 0.1: \[ x^2 = 100 \] Taking the square root: \[ x = \pm 10 \]

3. Equation: \(6v^2 + 24 = 0\)

   Rearranging gives: \[ 6v^2 = -24 \] Dividing by 6: \[ v^2 = -4 \] This does not yield real solutions because the square of a number cannot be negative. The solutions are: \[ v = \pm 2i \]

4. Equation: \(-x^2 + 3 = 0\)

   Rearranging gives: \[ x^2 = 3 \] Taking the square root: \[ x = \pm \sqrt{3} \]

5. Equation: \(y^2 - \frac{1}{9} = 0\)

   Rearranging gives: \[ y^2 = \frac{1}{9} \] Taking the square root: \[ y = \pm \frac{1}{3} \]

6. Equation: \(3m^2 - 1 = 0\)

   Rearranging gives: \[ 3m^2 = 1 \] Dividing by 3: \[ m^2 = \frac{1}{3} \] Taking the square root: \[ m = \pm \frac{1}{\sqrt{3}} \quad \text{or} \quad m = \pm \frac{\sqrt{3}}{3} \]

Summary of Solutions:

1. \( x = \pm \frac{3}{2} \)
2. \( x = \pm 10 \)
3. \( v = \pm 2i \) (complex solutions)
4. \( x = \pm \sqrt{3} \)
5. \( y = \pm \frac{1}{3} \)
6. \( m = \pm \frac{\sqrt{3}}{3} \)