# Let `f(x) = x^2+2x+9 ` where `x inRR` (1) If `alpha, beta` are the roots of f(x)=0,obtain the quadratics equation whose roots are `alpha^2-1` and `beta^2-1` .

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### 1 Answer

`alpha and beta ` are roots of equation `x^2+2x+9=0`

Thus

`alpha+beta=-2 `

`and`

`alpha.beta=9`

We wish to find an equation whose roots are

`alpha^2-1 and beta^2-1`

i.e

`(x-alpha^2+1)(x-beta^2+1)=0`

`x^2-(alpha^2+beta^2-2)x+(alpha^2-1)(beta^2-1)=0` (i)

But

`alpha^2+beta^2-2=(alpha+beta)^2-2alpha beta -2`

`=(-2)^2-2xx9-2=4-18-2=-16`

and

`(alpha^2-1)(beta^2-1)=alpha^2beta^2-(alpha^2+beta^2)+1`

`=9^2-(-14)+1=81+14+1=96`

substitute these values in (i),we get required solution as

`x^2+16x+96=0`