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

Question

`(a)/(a - x) + (b)/(b - x) = ((a + b)^(2) )/(ab)`
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Answer 1

`x_1=(a(a^2+ab+b^2))/((a+b)^2)\\ \\x_2=(b(a^2+ab+b^2))/((a+b)^2)`

Explanation:

First, simplify the left part:

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

Now, cross multiply:

`ab\cdot (ab-ax+ab-bx)=(a+b)^2\cdot (ab-ax-bx+x^2)\\ \\2(ab)^2-a^2bx-ab^2x=(a^2+2ab+b^2)(ab-ax-bx+x^2)\\ \\2(ab)^2-a^2bx-ab^2x=a^3b-a^3x-a^2bx+a^2x^2+2(ab)^2-2a^2bx-2ab^2x+2abx^2+ab^3-ab^2x-b^3x+b^2x^2\\ \\a^3b-a^3x+a^2x^2-2a^2bx-2ab^2x+2abx^2+ab^3-b^3x+b^2x^2=0\\ \\x^2(a^2+2ab+b^2)+x(-a^3-2a^2b-2ab^2-b^3)+a^3b+ab^3=0\\ \\(a+b)^2x^2-x((a+b)(a^2-ab+b^2)+2ab(a+b))+ab(a^2+b^2)=0\\ \\(a+b)^2x^2-x((a+b)(a^2+ab+b^2))+ab(a^2+b^2)=0\\ \\(a+b)^2x^2-x(a+b)(a^2+ab+b^2)+ab(a^2+b^2)=0`

`D=(a+b)^2(a^2+ab+b^2)^2-4(a+b)^2ab(a^2+b^2)\\ \\=(a+b)^2(a^4+(ab)^2+b^4+2a^3b+2a^2b^2+2ab^3-4a^3b-4ab^3)\\ \\=(a+b)^2(a^2-ab+b^2)^2\\ \\√(D)=(a+b)(a^2-ab+b^2)=a^3-b^3`

So,

`x_(1,2)=((a+b)(a^2+ab+b^2)\pm (a^3-b^3))/(2(a+b)^2)\\ \\x_1=(a^3+a^2b+ab^2+ba^2+ab^2+b^3+a^3-b^3)/(2(a+b)^2)=(2a^3+2a^2b+2ab^2)/(2(a+b)^2)=(2a(a^2+ab+b^2))/(2(a+b)^2)=(a(a^2+ab+b^2))/((a+b)^2)\\ \\x_2=(a^3+a^2b+ab^2+ba^2+ab^2+b^3-a^3+b^3)/(2(a+b)^2)=(2b^3+2a^2b+2ab^2)/(2(a+b)^2)=(2b(a^2+ab+b^2))/(2(a+b)^2)=(b(a^2+ab+b^2))/((a+b)^2)`