y=x^2 +x , y=2x^2 +x+m and y=kx+b if one point intersects, find k∙b=?


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Question Number 146611 by mathdanisur last updated on 14/Jul/21

$y = x^{2} + x, y = 2 x^{2} + x + m a n d y = k x + b$ $i f o n e p o i n t i n t e r s e c t s, f i n d k \cdot b = ?$
	Answered by mr W last updated on 14/Jul/21

	$y = x^{2} + x = k x + b$ $x^{2} + (1 - k) x - b = 0$ $Δ = {(1 - k)}^{2} + 4 b = 0$ $y = 2 x^{2} + x + m = k x + b$ $2 x^{2} + (1 - k) x + m - b = 0$ $Δ = {(1 - k)}^{2} + 8 (b - m) = 0$ $4 b = 8 (b - m)$ $\Rightarrow b = 2 m$ ${(1 - k)}^{2} = - 8 m$ $1 - k = \pm 2 \sqrt{- 2 m}$ $k = - 1 \pm 2 \sqrt{- 2 m}$ $\Rightarrow k b = 2 m (- 1 \pm 2 \sqrt{- 2 m})$
	Commented by mathdanisur last updated on 14/Jul/21

	$T h a n k y o u S e r, c o o l$
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