Introduction:
The given equation is 7√x÷1-x 8√1-x÷x = 15. We need to find the roots of the equation.

Method:
To solve the given equation, we will use the following steps:

Step 1: Simplify the given equation
Step 2: Rewrite the equation in the standard form
Step 3: Factorize the equation
Step 4: Find the roots of the equation

Step 1: Simplify the given equation
We can simplify the given equation by dividing both sides by 15.

7√x÷1-x 8√1-x÷x = 15
=> (7/15)√x÷1-x (8/15)√1-x÷x = 1

Step 2: Rewrite the equation in the standard form
To rewrite the equation in the standard form, we can assume y = √x÷1-x and z = √1-x÷x. Then, the given equation becomes:

(7/15)y + (8/15)z = 1

Step 3: Factorize the equation
To factorize the equation, we can use the formula for the sum of two fractions:

(a/b) + (c/d) = (ad + bc)/bd

Using this formula, we can write:

(7/15)y + (8/15)z = (7y + 8z)/15

Therefore, the given equation becomes:

(7y + 8z)/15 = 1

Multiplying both sides by 15, we get:

7y + 8z = 15

Step 4: Find the roots of the equation
To find the roots of the equation, we can use the substitution method. From the assumption we made earlier, we know that y = √x÷1-x and z = √1-x÷x. Using these values, we can find the roots of the equation.

Substituting y = √x÷1-x and z = √1-x÷x in the equation 7y + 8z = 15, we get:

7(√x÷1-x) + 8(√1-x÷x) = 15

Squaring both sides, we get:

49x÷(1-x) + 112 + 64(1-x)÷x = 225

Multiplying both sides by x(1-x), we get:

49x + 112x(1-x) + 64(1-x) = 225x(1-x)

Simplifying the equation, we get:

64x² - 113x + 64 = 0

Using the quadratic formula, we can find the roots of the equation:

x = 1/64 or x = 64/49

Therefore, the roots of the equation are x = 1/64 and x = 64/49.

Conclusion:
In this way, we can find the roots of the given equation using the substitution method.