One line has a positive slope and the other a negative slope, they are bound to intercept eventually, so that's why there can't be zero solutions. Two lines can't intercept each other twice, and if they have opposite slopes they can't have infinite solutions.
let;
First quadratic equation whose graph open upward is
[tex]ax^{2} + bx + c = 0[/tex] , where (a≠0)
when a>0 is positive the graph upward .
And second quadratic equation whose graph open downwards is
[tex]-ax^{2} + bx + c = 0[/tex] , where ( a≠0 )
When a<0 is negative the graph open downwards.
There are 3 possibilities for the graph as shown in the figure.
- Having zero intersection where the graph aren't touching.
- Having one intersection point where the graph touching at one point.
- Having two intersection points where the graph overlap.
So, the greatest possible number of intersections for these graphs is 2 as shown in figure.
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