Author Topic: question help discriminant? (Read 510 times) Tweet Share

Yoda · « **on:** June 02, 2014, 05:39:51 pm »

The equations y=mx and y=x^2(x-k) intersect at the point (0,0). Find the relationship between m and k (hint use the discriminant)

Zealous · « **Reply #1 on:** June 02, 2014, 06:02:19 pm »

Whenever we want to find the intersection of two graphs, we must first equate them and solve for x:

$\begin{eqnarray*}\\\text{{Equate\ Graphs:}}\\mx & = & x^{2}(x-k)\implies x^{3}-kx^{2}-mx=0\implies x(x^{2}-kx-m)=0\\\text{{Null\ Factor\ Law:}}\ & \implies & x=0\ \text{{or}}\ x^{2}-kx-m=0\\\end{eqnarray*}$

So we have found our first intersection point when x=0. The question states that there is only 1 intersection point, so there must be no solution to $x^2-kx-m=0$ (or else there will be more intersection points), hence the discrimant must be less than 0.

$\\\begin{eqnarray*}\\\triangle=b^{2}-4ac & < & 0\\k^{2}-4(1)(-m) & < & 0\\k^{2} & < & -4m\\m & < & -\frac{k^{2}}{4}\\\end{eqnarray*}$

(assuming m and k are positive!)

Test it out for yourself! as long as you put in an m value which fulfills the condition, you'll only ever get one intersection point of between the two graphs.

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Author Topic: question help discriminant? (Read 510 times) Tweet Share

Yoda

question help discriminant?

Zealous

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