I have been studying on cubic equations for a while and see that the cubic equation needs to be in t

Erick Clay 2022-05-29 Answered
I have been studying on cubic equations for a while and see that the cubic equation needs to be in the form of m x 3 + p x + q = 0 so that we can find the roots easily. In order to obtain such an equation having no quadratic term for a cubic equation in the form of x 3 + a x 2 + b x + c = 0, x value needs to be replaced with t a 3 .
I realised that the second derivative of any cubic equation in the form of x 3 + a x 2 + b x + c = 0 is y = 6 x + 2 a, and when we equalize y to 0, x is equal to a 3 . This had me thinking about any possible relation between the quadratic term and the second derivative.
It is also sort of the same in quadratic functions in the form of x 2 + a x + b = 0. When we replace x with a 2 , which is the first derivative of a quadratic function and the vertex point of it, we obtain the vertex form of the equation which does not have the linear term, x 1 .
Is there a relation between the quadratic term and the second derivative of a cubic equation? If the answer is yes, what is it and how is it observed on graphs, in equations?
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Answers (2)

mnaonavl
Answered 2022-05-30 Author has 7 answers
If you look at the inflection point of
y = x 3 + a x 2 + b x + c
where the second derivative is 0, you realize that it happens at x = a / 3 while the inflection point of
m x 3 + p x + q
happens at x = 0
Thus the transformation is moving the inflection point to x = 0
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Bailee Landry
Answered 2022-05-31 Author has 3 answers
There is not one relation, but two.
You noticed the first one: y is equal to zero exactly once, and this happens at the centre of symmetry of the curve, which is also its only inflection point. If you look at the abscissa ( x-value) of this point, it will tell you what a is (by the formula that you obtained).
The second relation is that in y = 6 x + 2 a, you can also equal not y , but y, to 0. This tells you that at the point where the curve intersects the y-axis, the curve is -shaped if a > 0, and -shaped if a < 0. if a = 0. then it is neither, because you have the inflection point right there.
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