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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.

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

Solve for the values of m that make the quadratic equation mx^2 - 4x + 2m = 0 have equal roots. Learn how to apply the discriminant condition and solve algebraic equations step-by-step, suitable for high school students.

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