Let's solve the equation provided in the image:

The equation is:

$25z^2 + 140 = -29$

Step 1: Simplify the equation

First, subtract 140 from both sides:

$25z^2 = -29 - 140$

$25z^2 = -169$

Step 2: Solve for $z^2$

Now, divide both sides by 25:

$z^2 = \frac{-169}{25}$

$z^2 = -\frac{169}{25}$

Step 3: Take the square root of both sides

To solve for $z$, take the square root of both sides:

$z = \pm \sqrt{-\frac{169}{25}}$

Since the square root of a negative number results in an imaginary number, we conclude that there are no real solutions.

Thus, the correct answer is:

• no real solutions.

Would you like further details or have any other questions? Here are some related questions for your understanding:

1. What is the difference between real and imaginary solutions in quadratic equations?
2. How do you handle negative numbers under square roots in equations?
3. What does the discriminant tell you about the nature of the roots in a quadratic equation?
4. How can we solve a quadratic equation when the discriminant is negative?
5. Can you solve the equation $9x^2 + 16 = -25$?

Tip: When you have a negative number under the square root, it suggests the solution involves imaginary numbers, which are based on $i$ (where $i = \sqrt{-1}$).