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Solve the following quadratic equation for x: x^2+(a/(a+b)+(a+b)/a)x+1=0 - Mathematics

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Solve the following quadratic equation for x:

`x^2+(a/(a+b)+(a+b)/a)x+1=0`

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Solution

Given: 

`x^2+(a/(a+b)+(a+b)/a)x+1=0`

Let `a/(a+b)`be t

Thus, the equation becomes 

`x^2+(t+1/t)x+1=0`

x2+(t+`1/t`)x+1=0

x2+(`(t^2+1)/1`)x+1=0

⇒tx2+(t2+1)x+t=0

tx2+t2x+x+t=0

(tx2+t2x)+(x+t)=0

tx(x+t)+1(x+t)=0

(tx+1)(x+t)=0

tx+1=0, x+t=0

`=>x=(-1)/t,x= -t`

`=>x = (-1)/(a/(a+b)), x = -a/(a+b)`

 `=>x= -(a+b)/a, x= -a/(a+b)`

Concept: Solutions of Quadratic Equations by Factorization
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