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To determine which equation has no real solution, we can analyze the discriminant of each quadratic equation. The discriminant of a quadratic equation \( ax^2 + bx + c = 0 \) is given by the formula:

\[
\Delta = b^2 - 4ac
\]

- If \(\Delta > 0\), there are two real solutions.
- If \(\Delta = 0\), there is one real solution.
- If \(\Delta < 0\), there are no real solutions.

Let’s go through each option:

1. \( 2x^2 + 2x + 15 = 0 \)

\[
a = 2, \, b = 2, \, c = 15
\]

Discriminant:

\[
\Delta = 2^2 - 4(2)(15) = 4 - 120 = -116
\]

Since \(\Delta < 0\), this equation has **no real solutions**.

2. \( 2x^2 + 5x - 3 = 0 \)

\[
a = 2, \, b = 5, \, c = -3
\]

Discriminant:

\[
\Delta = 5^2 - 4(2)(-3) = 25 + 24 = 49
\]

Since \(\Delta > 0\), this equation has two real solutions.

3. \( x^2 + 7x + 2 = 0 \)

\[
a = 1, \, b = 7, \, c = 2
\]

Discriminant:

\[
\Delta = 7^2 - 4(1)(2) = 49 - 8 = 41
\]

Since \(\Delta > 0\), this equation has two real solutions.

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

\[
a = 1, \, b = -4, \, c = 2
\]

Discriminant:

\[
\Delta = (-4)^2 - 4(1)(2) = 16 - 8 = 8
\]

Since \(\Delta > 0\), this equation has two real solutions.

### Conclusion:
The first equation, \( 2x^2 + 2x + 15 = 0 \), has no real solution.

Would you like any more details or have any questions?

Here are 5 related questions to expand your understanding:
1. How does the discriminant help determine the nature of roots?
2. Can a quadratic equation with complex roots ever have real solutions?
3. What is the geometric interpretation of an equation with no real solutions?
4. How can you solve a quadratic equation when the discriminant is negative?
5. What is the significance of the sign of the discriminant in real-world problems?

**Tip:** Always check the discriminant first when solving quadratic equations to quickly determine the nature of the solutions!

Find out which quadratic equation has no real solution by analyzing the discriminant. This exercise explains the discriminant formula and helps identify when a quadratic equation lacks real solutions. Suitable for students in Grades 8-10.

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