If {x^2} + 2x + 2xy + my - 3 has two rational | vedclass

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(C) The given expression is $f(x, y) = x^2 + 2x(1 + y) + (my - 3)$.For the expression to have rational factors,its discriminant $D$ with respect to $x

Vedclass · Accepted Answer

$6, -2$

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(C) The given expression is $f(x, y) = x^2 + 2x(1 + y) + (my - 3)$.For the expression to have rational factors,its discriminant $D$ with respect to $x$ must be a perfect square of a linear polynomial in $y$.$D = [2(1 + y)]^2 - 4(1)(my - 3) = 4(1 + 2y + y^2 - my + 3) = 4(y^2 + y(2 - m) + 4)$.For $D$ to be a perfect square,the quadratic $y^2 + y(2 - m) + 4$ must be a perfect square,which means its discriminant must be zero.$D_y = (2 - m)^2 - 4(1)(4) = 0$.$(2 - m)^2 = 16$.$2 - m = \pm 4$.If $2 - m = 4$,then $m = -2$.If $2 - m = -4$,then $m = 6$.Thus,the values of $m$ are $6$ and $-2$.

If ${x^2} + 2x + 2xy + my - 3$ has two rational factors,then the value of $m$ will be

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