https://file.aiquickdraw.com/mathbot/file/17289765626817c94v0b0.jpg

The inequality shown in the image is:

\[
2 + x^2 > 2x
\]

To solve this inequality, let's rearrange it to bring all terms to one side:

\[
x^2 - 2x + 2 > 0
\]

Now, we will analyze the quadratic expression \( x^2 - 2x + 2 \) to see when it is positive.

1. **Discriminant Check**: The discriminant of the quadratic equation \( ax^2 + bx + c = 0 \) is given by \( \Delta = b^2 - 4ac \). Here:
   - \( a = 1 \)
   - \( b = -2 \)
   - \( c = 2 \)

\[
\Delta = (-2)^2 - 4(1)(2) = 4 - 8 = -4
\]

Since the discriminant is negative (\( \Delta < 0 \)), the quadratic has no real roots. This means that the parabola does not intersect the x-axis and is always above the x-axis (because the coefficient of \( x^2 \), \( a = 1 \), is positive, which means the parabola opens upwards).

2. **Conclusion**: Since \( x^2 - 2x + 2 > 0 \) for all real values of \( x \), the inequality is always true.

Thus, the solution to the inequality \( 2 + x^2 > 2x \) is:

\[
\boxed{\text{All real numbers}}
\]

Would you like further details or have any questions on this? 

Here are 5 related questions you could explore:
1. What is the role of the discriminant in solving quadratic equations and inequalities?
2. How can we determine whether a quadratic expression is always positive or negative?
3. What are the different methods for solving quadratic inequalities?
4. How do you interpret the graph of \( x^2 - 2x + 2 \) and what does it reveal about the inequality?
5. Could this inequality have a different solution if the sign were flipped (i.e., \( 2 + x^2 < 2x \))?

**Tip**: When solving quadratic inequalities, always check the sign of the discriminant first—it tells you whether the quadratic will cross the x-axis or remain entirely above or below it.

Discover how to solve the quadratic inequality 2 + x^2 > 2x. This problem involves using the discriminant to determine the nature of the quadratic and analyze when it is always positive. Ideal for students in Grades 9-12.

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