# If $\alpha ,\beta$ are roots of $x^{2}+x+2= 0$, then the equation whose roots are $\alpha ^{2}+1$ & $\beta ^{2}+1$ is Option 1) $x^{2}-x-2= 0$ Option 2) $x^{2}-x+2= 0$ Option 3) $x^{2}+x-2= 0$ Option 4) $x^{2}+x+2= 0$

S subam

$\\*S=(\alpha ^{2}+1)+(\beta ^{2}+1)=(\alpha +\beta )^{2}-2\alpha \beta +2\\*S=1-2(2)+2=-1\Rightarrow S=-1\\*P=(\alpha ^{2}+1)(\beta ^{2}+1)=(\alpha \beta )^{2}+(\alpha ^{2}+\beta ^{2})+1\\*P=(\alpha \beta )^{2}+(\alpha +\beta )^{2}-2\alpha \beta +1\\*P=4+1-2(2)+1=2\Rightarrow P=2\\*\therefore Equation\; :\, \; x^{2}+x+2=0$

To form a Quadratic Equation given the roots -

$x^{2}-Sx+P= 0$

- wherein

S = Sum of roots

P = Product of roots

Option 1)

$x^{2}-x-2= 0$

This is incorrect

Option 2)

$x^{2}-x+2= 0$

This is incorrect

Option 3)

$x^{2}+x-2= 0$

This is incorrect

Option 4)

$x^{2}+x+2= 0$

This is correct

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