The given equation is:

$x^2 - xy^2 = 2$

We need to determine which of the provided points satisfy this equation. Let's check each option by substituting the coordinates of the points into the equation.

1. Option A: $\left( -\sqrt{2}, 0 \right)$

Substitute $x = -\sqrt{2}$ and $y = 0$ into the equation:

$(-\sqrt{2})^2 - (-\sqrt{2})(0)^2 = 2 \Rightarrow 2 - 0 = 2$

This is true, so Option A satisfies the equation.

2. Option B: $\left( 0, \sqrt{2} \right)$

Substitute $x = 0$ and $y = \sqrt{2}$ into the equation:

$0^2 - 0(\sqrt{2})^2 = 2 \Rightarrow 0 - 0 = 2$

This is false, so Option B does not satisfy the equation.

3. Option C: $\left( 0, 0 \right)$

Substitute $x = 0$ and $y = 0$ into the equation:

$0^2 - 0(0)^2 = 2 \Rightarrow 0 - 0 = 2$

This is false, so Option C does not satisfy the equation.

4. Option D: $\left( \frac{\sqrt{2}}{2}, (4.5)^{1/4} \right)$

Substitute $x = \frac{\sqrt{2}}{2}$ and $y = (4.5)^{1/4}$ into the equation:

$\left( \frac{\sqrt{2}}{2} \right)^2 - \left( \frac{\sqrt{2}}{2} \right) \cdot \left( (4.5)^{1/4} \right)^2 = 2$

We would need to calculate:

$\left( \frac{\sqrt{2}}{2} \right)^2 = \frac{2}{4} = \frac{1}{2}$ $(4.5)^{1/4} = \sqrt{\sqrt{4.5}} = 1.2757$ $\left( 1.2757 \right)^2 = 1.6274$ So: $\frac{1}{2} - \left( \frac{\sqrt{2}}{2} \cdot 1.6274 \right) = 2$

But this does not result in 2, so Option D does not satisfy the equation.

Conclusion:

Only Option A satisfies the equation $x^2 - xy^2 = 2$.

Would you like further clarification or details on any of the steps?

Here are five related questions to deepen your understanding:

1. How do we verify solutions to an equation involving two variables?
2. What is the significance of squaring numbers like $\sqrt{2}$ in equations?
3. What mathematical tools help simplify complex square root expressions?
4. Why do certain points satisfy an equation while others don't?
5. How can substitution be used effectively in solving multivariable equations?

Tip: When verifying solutions to an equation, substitute the values carefully and simplify step by step to avoid errors.