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To determine which quadratic equation has no real solutions, we use the **discriminant** of the quadratic equation.

The discriminant for a quadratic equation \( ax^2 + bx + c = 0 \) is given by:

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

- If \( \Delta > 0 \), the equation has two real solutions.
- If \( \Delta = 0 \), the equation has exactly one real solution.
- If \( \Delta < 0 \), the equation has no real solutions.

Now let's calculate the discriminant for each option:

### Option A: \( x^2 + 14x - 49 = 0 \)
- \( a = 1 \), \( b = 14 \), \( c = -49 \)
\[
\Delta = 14^2 - 4(1)(-49) = 196 + 196 = 392
\]
Since \( \Delta > 0 \), it has two real solutions.

### Option B: \( x^2 - 14x + 49 = 0 \)
- \( a = 1 \), \( b = -14 \), \( c = 49 \)
\[
\Delta = (-14)^2 - 4(1)(49) = 196 - 196 = 0
\]
Since \( \Delta = 0 \), it has exactly one real solution.

### Option C: \( 5x^2 - 14x - 49 = 0 \)
- \( a = 5 \), \( b = -14 \), \( c = -49 \)
\[
\Delta = (-14)^2 - 4(5)(-49) = 196 + 980 = 1176
\]
Since \( \Delta > 0 \), it has two real solutions.

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

### Conclusion:
The quadratic equation in **Option D** has no real solutions.

Would you like further details on the discriminant or another topic?

Here are 5 related questions to explore:
1. How do you find the roots of a quadratic equation using the quadratic formula?
2. What happens if the discriminant is a perfect square?
3. How can the vertex form of a quadratic equation help determine the number of solutions?
4. How do complex solutions arise from a quadratic equation?
5. What role does the discriminant play in determining the shape of a parabola?

**Tip:** Always check the discriminant first when analyzing the nature of the solutions of a quadratic equation.

Which quadratic equation has no real solutions?

Learn how to determine which quadratic equation has no real solutions by using the discriminant formula. This problem focuses on analyzing the discriminant to identify equations with real or complex roots. Suitable for students in Grades 8-10.

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