If `alpha`   , `beta` are the roots of the equation `x^2-4x-3 =0` , then `alpha^2+alpha*beta+ beta^2=` A. −13 B. 5 C. 13 D. 16 E. 19 `x^2-4x-3 = 0`

The roots of the above quadratic function is `alpha` and `beta` .

Then we know that;

`alpha+beta = -(-4)/1 = 4`

`alphaxxbeta = -3/1 = -3`

`alpha^2+alphaxxbeta+beta^2`

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

`(x+y)^2 = x^2+2xy+y^2`

`(x^2+y^2) = (x+y)^2-2xy`

`alpha^2+alphaxxbeta+beta^2`

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

`= 4^2-(-3)`

`= 19`

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`x^2-4x-3 = 0`

The roots of the above quadratic function is `alpha` and `beta` .

Then we know that;

`alpha+beta = -(-4)/1 = 4`

`alphaxxbeta = -3/1 = -3`

`alpha^2+alphaxxbeta+beta^2`

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

`(x+y)^2 = x^2+2xy+y^2`

`(x^2+y^2) = (x+y)^2-2xy`

`alpha^2+alphaxxbeta+beta^2`

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

`= 4^2-(-3)`

`= 19`

So the answer is 19 and it is at E)