To solve the equation \( x^2 = \frac{2564}{25} \), we first simplify the fraction:
\[ \frac{2564}{25} = 102.56 \]
Next, we can rewrite the equation as:
\[ x^2 = 102.56 \]
Taking the square root of both sides gives us two possible solutions:
\[ x = \sqrt{102.56} \quad \text{and} \quad x = -\sqrt{102.56} \]
Calculating the square root:
\[ \sqrt{102.56} = 10.126 \]
Thus, the two solutions are:
\[ x \approx 10.126 \quad \text{and} \quad x \approx -10.126 \]
To express these in simplest fraction form, we can recognize that:
\[ \sqrt{\frac{2564}{25}} = \frac{\sqrt{2564}}{\sqrt{25}} = \frac{\sqrt{2564}}{5} \]
Now, let's simplify \( \sqrt{2564} \):
2564 can be factored into \( 4 \times 641 \), so:
\[ \sqrt{2564} = \sqrt{4 \times 641} = 2\sqrt{641} \]
Putting this back into our expression gives:
\[ x = \frac{2\sqrt{641}}{5} \quad \text{and} \quad x = -\frac{2\sqrt{641}}{5} \]
Therefore, the two real solutions for the equation \( x^2 = \frac{2564}{25} \) in simplest fraction form are:
\[ -\frac{2\sqrt{641}}{5}, \frac{2\sqrt{641}}{5} \]
You can write them in order as:
\[ \boxed{-\frac{2\sqrt{641}}{5}, \frac{2\sqrt{641}}{5}} \]