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It needs to be pointed out: Here we have a special quadratic expression. The quadratic formula is similar, but uses b2 – 4ac rather than b2 – 4bc. Let's see how the quadratic formula applies here. Let`s present the same data differently to reach the same conclusion.
Here, we have written the expression as y2 times a x over y2 plus b x over y + c. Since x and y are both non-negative, the expression will never be non-negative.
This depends on the ratio between x and y, which means it depends on which direction you're moving away from the origin--the point (0, 0). If a discriminant is positive, then it means that this equation has a solution.
One way to convince yourself that a quantity can be both positive and negative is to plot its square root. The square root might look like this, or it might look like that, depending on the sign of a.
But, in either case, it will take values of both signs. So that means we have a saddle point.
While the overall situation is indicated by the quadratic equation b squared minus 4ac, this equation never takes the value 0 because if b squared minus 4ac is negative, then that means there's no solution for the quadratic equation and thus no minimum or maximum. And we know that a x over y squared plus b x over y plus c is either always positive or always negative depending on the sign of a, which tells us that our function will be either always positive or always negative.
We'll have a maximum or a minimum, depending on whether the discriminant is positive or negative. This will tell us whether this quadratic quantity has always the same sign or whether I can cross the value zero when you have a root of a quadratic.
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