You have provided a list of problems involving quadratic equations and their properties. Let's tackle one of them:

"Find the values of parameter $m$ for which the quadratic equation $mx^2 - 4x + 2m = 0$ has equal roots."

A quadratic equation $ax^2 + bx + c = 0$ has equal roots if the discriminant $b^2 - 4ac = 0$. For the equation $mx^2 - 4x + 2m = 0$:

• $a = m$
• $b = -4$
• $c = 2m$

The discriminant equation is: $(-4)^2 - 4 \cdot m \cdot 2m = 0$

We can solve this to find $m$. Let's do that.The values of the parameter $m$ for which the quadratic equation $mx^2 - 4x + 2m = 0$ has equal roots are $m = -\sqrt{2}$ and $m = \sqrt{2}$.

Would you like to explore another problem or need more details on this one?

Here are some related questions you might consider:

1. What would be the roots of the equation $mx^2 - 4x + 2m = 0$ when $m = -\sqrt{2}$ or $m = \sqrt{2}$?
2. How does the value of $m$ affect the vertex of the parabola represented by $mx^2 - 4x + 2m$?
3. Could you solve a similar problem with different coefficients?
4. What would be the graph of $mx^2 - 4x + 2m = 0$ look like for different values of $m$?
5. Can you explain why a quadratic equation might have no real roots?

Tip: When checking for the discriminant in quadratic equations, remember that a positive discriminant indicates two distinct real roots, zero indicates a repeated root, and a negative discriminant indicates no real roots.