# Depressing a Cubic Equation

Suppose I have a cubic equation of $$x^3 + ax^2 + bx + c=0.$$ What steps would one take to eliminate the $x^2$ term? Given an elliptic curve that is not of the form $$Y^2 = X^3 + AX+B,$$ my goal would be to normalize the elliptic curve to that form with the appropriate substitutions. Handling the $Y$ side isn't a problem as all that is needed is to complete the square, but I am not sure how to get rid of the $x^2$ term on the $X$ side.

I'm not sure what subject this falls under so additional tags are welcomed.

• Show her the morning newspaper, that should work!
– Pedro
Commented Jul 31, 2013 at 0:11
• @PeterTamaroff Joke not understood. :(
– mick
Commented Jul 31, 2013 at 1:41

Take $$x\mapsto x-\frac{a}{3}$$ to eliminate the $x^2$ term, since $${\left( {x - \frac{a}{3}} \right)^3} = {x^3} - a{x^2} + \frac{{x{a^2}}}{3} - \frac{{{a^3}}}{{27}}$$

• I am doing an exercise related to this and will remember it forever now :). $-\dfrac a3$ is the point of inflection of the given cubic, so doing said substitution shifts the graph so that the point of inflection lies on the $y$ axis. After that, by symmetry we know $f'(x) = f'(-x)$, which means the derivative must be a parabola of the form $x^2 + \gamma$, which means that $f$ must be a depressed cubic.
– Ovi
Commented Feb 24, 2017 at 14:48

This is my answer for the general case of $$ax^3+bx^2+cx+d$$

So I'll take that $$a=1$$ for clean computation. I'm sure that's an easy step. Just divide the whole thing by a for any $$a \ge 1$$.

So we're left with $$x^3+bx^2+cx+d$$

So in order to get rid of the $$x^2$$ term you want to rewrite the thing in terms of another variable which achieves this purpose.

Let's say $$y$$. We rewrite in terms of $$y$$. Therefore $$x=y+z$$ for a constant $$z$$. How do we pick out $$z$$?

$$(y+z)^3+b(y+z)^2+c(y+z)+d$$

$$y^3+z^3+3yz(y+z)+b(y^2+2yz+z^2)+cy+cz+d$$

All very nice and fun to expand but I'm interested in the terms with $$y^2$$ only.

$$3zy^2+by^2$$

$$(3z+b)y^2$$

So if we need that term to cease to exist, then ensure that coefficient in brackets is 0.

$$3z+b=0$$

$$z=-\dfrac{b}{3}$$

There we have it. To depress a cubic you rewrite x in terms of another variable $$y\text{(or whatever you want to call it)}$$ where $$x=y-\dfrac{b}{3}$$ and b is the original $$x^2$$ term.

Could someone tell me what Q.E.D means so I can be using it in future answers?