Solve for x: (x-a)/b +(x-b)/a = b/(x-a) +a/(x-b)
To solve the equation (x-a)/b +(x-b)/a = b/(x-a) +a/(x-b) for x, we can follow the steps below:
Step 1: Simplify the equation by finding a common denominator for all the fractions involved. In this case, the common denominator is ab(x-a)(x-b).
Multiplying each term in the equation by ab(x-a)(x-b), we have:
a(x-b)(x-a) + b(x-a)(x-b) = b^2 + a^2
Step 2: Expand and simplify the equation:
a(x^2 - ax - bx + ab) + b(x^2 - ax - bx + ab) = b^2 + a^2
Expanding the equation further:
ax^2 - a^2x - abx + a^2 + bx^2 - abx - b^2x + ab^2 = b^2 + a^2
Combining like terms:
ax^2 + bx^2 - 2abx + 2a^2 - 2b^2x + ab^2 = b^2 + a^2
Step 3: Rearrange the equation to form a quadratic equation:
(ax^2 + bx^2) - (2abx + 2b^2x) + (2a^2 - ab^2) = b^2 + a^2
Combine the like terms:
(a + b)x^2 - (2ab + 2b^2)x + (2a^2 - ab^2) = b^2 + a^2
Step 4: Set the equation equal to zero by subtracting (b^2 + a^2) from both sides:
(a + b)x^2 - (2ab + 2b^2)x + (2a^2 - ab^2) - (b^2 + a^2) = 0
Simplifying further:
(a + b)x^2 - (2ab + 2b^2)x + (2a^2 - ab^2 - b^2 - a^2) = 0
(a + b)x^2 - (2ab + 2b^2)x + (a^2 - ab^2 - b^2) = 0
Step 5: Now, we can solve this quadratic equation. We can use the quadratic formula: x = (-b ± √(b^2 - 4ac)) / (2a). In our case, a = (a + b), b = -(2ab + 2b^2), and c = (a^2 - ab^2 - b^2).
Substituting the values into the quadratic formula:
x = ( -(-(2ab + 2b^2)) ± √((2ab + 2b^2)^2 - 4(a + b)(a^2 - ab^2 - b^2))) / (2(a + b))
Simplifying further:
x = (2ab + 2b^2 ± √(4a^2b^2 + 8ab^3 + 4b^4 - 4a^3 - 4a^2b + 4ab^3 + 4b^2 - 4ab^2 - 4b^3)) / (2(a + b))
Step 6: Continue simplifying the expression under the square root:
x = (2ab + 2b^2 ± √(4a^2b^2 - 4a^3 - 4a^2b - 4ab^2 + 4b^4 - 4b^3 + 4ab^3 + 4b^2)) / (2(a + b))
x = (2ab + 2b^2 ± √(-4a^3 + 4a^2b + 4ab^3 - 4b^3 - 4ab^2 + 4b^4 + 4a^2b^2 + 4b^2)) / (2(a + b))
x = (2ab + 2b^2 ± √(4a^2b^2 + 4ab^3 - 4a^3 - 4ab^2 - 4b^3 + 4b^4 + 4b^2)) / (2(a + b))
x = (2ab + 2b^2 ± 2√(a^2b^2 + ab^3 - a^3 - ab^2 - b^3 + b^4 + b^2)) / (2(a + b))
Step 7: Simplify the expression within the square root further:
x = (ab + b^2 ± √(a^2b^2 + ab^3 - a^3 - ab^2 - b^3 + b^4 + b^2)) / (a + b)
Now, we have found the general solution for x in terms of a, b, ab, and b^2.