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To determine which quadratic equation has no real solutions, we should evaluate the discriminant of each option. For a quadratic equation in the form \( ax^2 + bx + c = 0 \), the discriminant is given by:

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

If \(\Delta < 0\), the equation has no real solutions.

Let’s examine each option:

1. **Option A:** \( x^2 + 14x - 49 = 0 \)
   - \( a = 1 \), \( b = 14 \), \( c = -49 \)
   - Discriminant: \( \Delta = 14^2 - 4 \cdot 1 \cdot (-49) = 196 + 196 = 392 \)
   - Since \(\Delta > 0\), this equation has real solutions.

2. **Option B:** \( x^2 + 14x + 49 = 0 \)
   - \( a = 1 \), \( b = 14 \), \( c = 49 \)
   - Discriminant: \( \Delta = 14^2 - 4 \cdot 1 \cdot 49 = 196 - 196 = 0 \)
   - Since \(\Delta = 0\), this equation has a real (repeated) solution.

3. **Option C:** \( 5x^2 - 14x - 49 = 0 \)
   - \( a = 5 \), \( b = -14 \), \( c = -49 \)
   - Discriminant: \( \Delta = (-14)^2 - 4 \cdot 5 \cdot (-49) = 196 + 980 = 1176 \)
   - Since \(\Delta > 0\), this equation has real solutions.

4. **Option D:** \( 5x^2 - 14x + 49 = 0 \)
   - \( a = 5 \), \( b = -14 \), \( c = 49 \)
   - Discriminant: \( \Delta = (-14)^2 - 4 \cdot 5 \cdot 49 = 196 - 980 = -784 \)
   - Since \(\Delta < 0\), this equation has no real solutions.

### Conclusion
The correct answer is **Option D** because it has a negative discriminant, indicating no real solutions.

Would you like further details on discriminants or have any other questions?

### Related Questions:
1. How do discriminants determine the nature of solutions in quadratic equations?
2. What are complex roots, and when do they occur in quadratic equations?
3. Can a quadratic equation have exactly one real solution?
4. What are the implications of a zero discriminant in quadratic equations?
5. How do we complete the square for a quadratic equation?

### Tip:
For quick determination of real solutions in quadratics, always calculate the discriminant first.

Which quadratic equation has no real solutions?

Explore how to determine if a quadratic equation has real solutions by calculating the discriminant. This problem covers essential algebraic concepts, including the discriminant formula and conditions for real and complex roots. Ideal for Grade 10 students.

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