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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`

⇒x^{2}+(t+`1/t`)x+1=0

⇒x^{2}+(`(t^2+1)/1`)x+1=0

⇒tx^{2}+(t^{2}+1)x+t=0

⇒tx^{2}+t^{2}x+x+t=0

⇒(tx^{2}+t^{2}x)+(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)`

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