To find the value of x in the given geometric progression (G.P.) with terms x, 10, and -25, we need to determine the common ratio (r) and then solve for x using the formula for G.P.

Understanding Geometric Progression (G.P.)
A geometric progression is a sequence of numbers in which each term after the first is obtained by multiplying the preceding term by a fixed, non-zero number called the common ratio.

Step 1: Finding the Common Ratio (r)
To find the common ratio (r), we divide any term of the G.P. by its preceding term. Let's use the second and first terms:
r = 10 / x

Step 2: Using the Formula for G.P.
The formula for the nth term of a G.P. is given by:
an = a * r^(n-1)

In this case, we have the third term (-25) and the second term (10). Substituting these values into the formula, we get:
-25 = 10 * r^(3-1)

Step 3: Solving for the Common Ratio (r)
Simplifying the equation, we have:
-25 = 10 * r^2

Dividing both sides by 10, we obtain:
-2.5 = r^2

Taking the square root of both sides, we get two possible values for r:
r = √(-2.5) or r = -√(-2.5)

Note: Since r cannot be a complex number in this case, we can discard the negative square root and consider the positive value.

Step 4: Finding the Value of x
Now that we have the common ratio (r = √(-2.5)), we can use it to find the value of x. Let's substitute the common ratio into the equation for the second term:
r = 10 / x

Simplifying the equation, we have:
√(-2.5) = 10 / x

Cross-multiplying, we get:
√(-2.5) * x = 10

Dividing both sides by √(-2.5), we arrive at:
x = 10 / √(-2.5)

Therefore, the value of x in the given G.P. is 10 / √(-2.5).