For what value of m can the equation y^2 + 2x | vedclass

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(B) The general equation of a second-degree curve is $ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0$.Comparing $y^2 + 2xy + 2x + my - 3 = 0$ with the standar

Vedclass · Accepted Answer

$-2$

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(B) The general equation of a second-degree curve is $ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0$.Comparing $y^2 + 2xy + 2x + my - 3 = 0$ with the standard form,we get:$a = 0, b = 1, h = 1, g = 1, f = m/2, c = -3$.For the equation to represent a pair of straight lines (linear factors),the determinant $\Delta$ must be zero:$\Delta = \begin{vmatrix} a & h & g \ h & b & f \ g & f & c \end{vmatrix} = 0$Substituting the values:$\begin{vmatrix} 0 & 1 & 1 \ 1 & 1 & m/2 \ 1 & m/2 & -3 \end{vmatrix} = 0$Expanding the determinant:$0(1(-3) - (m/2)^2) - 1(1(-3) - 1(m/2)) + 1(1(m/2) - 1(1)) = 0$$0 - (-3 - m/2) + (m/2 - 1) = 0$$3 + m/2 + m/2 - 1 = 0$$m + 2 = 0$$m = -2$

For what value of $m$ can the equation $y^2 + 2xy + 2x + my - 3 = 0$ be factored into two linear factors?

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