Math, asked by itgoesfastermrbeast, 11 days ago

solve for x
 \frac{a}{x - a}  +  \frac{b}{x - b}  =  \frac{2c}{x - c}  \:
x≠ a,b,c
find roots of x​

Answers

Answered by mathdude500
4

Answer:

\qquad\qquad\boxed{ \sf{ \:\bf \: x =  0, \: \dfrac{ac +  cb  -   2ab}{2c - a - b} \: }} \\  \\

Step-by-step explanation:

Given equation is

\sf \: \dfrac{a}{x - a} +  \dfrac{b}{x - b} =  \dfrac{2c}{x - c}  \\  \\

\sf \: \dfrac{a(x - b) + b(x - a)}{(x - a)(x - b)} =  \dfrac{2c}{x - c}  \\  \\

\sf \: \dfrac{ax - ab + bx - ab}{ {x}^{2}  - bx - ax + ab} =  \dfrac{2c}{x - c}  \\  \\

\sf \: \dfrac{ax + bx - 2ab}{ {x}^{2}  - bx - ax + ab} =  \dfrac{2c}{x - c}  \\  \\

\sf \:2c ( {x}^{2} - ax - bx + ab) = (x - c)(ax + bx - 2ab) \\  \\

\sf \:2c {x}^{2} - 2acx - 2cbx + 2abc = a {x}^{2}  + b {x}^{2} - 2abx - acx - bcx + 2abc\\  \\

\sf \:2c {x}^{2} - 2acx - 2cbx  = a {x}^{2}  + b {x}^{2} - 2abx - acx - bcx \\  \\

\sf \:2c {x}^{2} - 2acx - 2cbx  -  a {x}^{2}  - b {x}^{2} +  2abx + acx  +  bcx = 0 \\  \\

\sf \:2c {x}^{2} - acx - cbx  -  a {x}^{2}  - b {x}^{2} +  2abx  = 0 \\  \\

\sf \:2c {x}^{2} -  {ax}^{2} -  {bx}^{2}   - acx - cbx  +  2abx  = 0 \\  \\

\sf \:x[ 2c x -  ax - bx - ac - cb  +  2ab]  = 0 \\  \\

\bf\implies\bf \:x  =  0 \\  \\

Or

\sf \: 2c x -  ax - bx - ac - cb  +  2ab = 0 \\  \\

\sf \: x(2c -  a - b) - ac - cb  +  2ab = 0 \\  \\

\sf \: x(2c -  a - b) = ac +  cb  -   2ab \\  \\

\bf\implies\bf \: x =  \dfrac{ac +  cb  -   2ab}{2c - a - b} \\  \\

Hence,

\bf\implies\bf \: x =  0, \: \dfrac{ac +  cb  -   2ab}{2c - a - b} \\  \\

\rule{190pt}{2pt}

Additional Information

Nature of roots :-

Let us consider a quadratic equation ax² + bx + c = 0, then nature of roots of quadratic equation depends upon Discriminant (D) of the quadratic equation.

If Discriminant, D > 0, then roots of the equation are real and unequal.

If Discriminant, D = 0, then roots of the equation are real and equal.

If Discriminant, D < 0, then roots of the equation are unreal or complex or imaginary.

Where,

Discriminant, D = b² - 4ac

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