Make x subject formula in (x/x+a) - (a/x+b)=1

Answer 1

To solve the equation:
(x/x+a) - (a/x+b) = 1

First, let's simplify the equation by multiplying both sides by the common denominator (x+a)(x+b):

((x)(x+b) - (a)(x+a)) / (x+a)(x+b) = (x+a)(x+b)

Expanding and simplifying the numerator:

(x^2 + xb - ax - a^2) / (x+a)(x+b) = (x^2 + ax + xb + ab)

Now, let's remove the denominators by multiplying both sides by (x+a)(x+b):

(x^2 + xb - ax - a^2) = (x^2 + ax + xb + ab)(x+a)(x+b)

Expanding and simplifying both sides:

x^2 + xb - ax - a^2 = (x^2 + ax + xb + ab)(x^2 + ax + a^2 + 2ab + b^2)

Now, let's continue simplifying by multiplying out the right side:

x^2 + xb - ax - a^2 = x^4 + 3ax^3 + 3a^2x^2 + 3abx^3 + 9a^2bx^2 + 6a^3bx + 3a^2b^2x + ab^2x^2 + 3a^3bx^2 + 9a^2b^2x + 6a^3bx + 2ab^2x + a^2b^2

Combining like terms:

x^2 + xb - ax - a^2 = x^4 + 3ax^3 + 3a^2x^2 + 3abx^3 + 9a^2bx^2 + 3a^2b^2x + 3a^3bx + ab^2x^2 + 3a^3bx^2 + 9a^2b^2x + 6a^3bx + 2ab^2x + a^2b^2

Now, let's move all the terms to one side of the equation:

x^4 + 3ax^3 + 3a^2x^2 + 3abx^3 + 9a^2bx^2 + 3a^2b^2x + 3a^3bx + ab^2x^2 + 3a^3bx^2 + 9a^2b^2x + 6a^3bx + 2ab^2x + a^2b^2 - x^2 - xb + ax + a^2 = 0

Rearranging the terms:

x^4 + (3a + 3b - 1)x^3 + (3a^2 + 9ab + ab^2 + 3a^3 + 3b^3 - 1)x^2 + (3a^2b + 3a^3b + 9a^2b^2 + 6a^3b + a + 2b)x + (a^2b^2 + a^2 - a) = 0

Therefore, the subject x is part of a quartic equation (a polynomial of degree 4):

x^4 + (3a + 3b - 1)x^3 + (3a^2 + 9ab + ab^2 + 3a^3 + 3b^3 - 1)x^2 + (3a^2b + 3a^3b + 9a^2b^2 + 6a^3b + a + 2b)x + (a^2b^2 + a^2 - a) = 0