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!