# Finding a cubic formula for roots of cubic equations

Solve for $$x$$, $$27x^3+21x+8=0$$

I would like to know if there exists an formula for cubic equations just like quadratic formula for quadratic equations.

• Cardano's formula works, but it is not exactly easy to use. Your example has a rational root, so it is best to start there. – lulu Feb 16 at 14:19
• Not clear what you want by way of more details, Robert. Could you be more explicit about what is missing in the answers that have been posted? – Gerry Myerson Feb 19 at 10:09
• Please, Robert, what more do you need? – Gerry Myerson Feb 21 at 0:33
• Sorry for delay in replying – user751264 Feb 21 at 16:56
• I am actually looking for a method which can be always applied (as the quadratic formula)to solve a cubic almost instantaneously – user751264 Feb 21 at 16:58

We can begin by making an general cubic equation into a depressed cubic( missing square term)

To make $$ax^3+bx^2+cx+d=0$$ a depressed cubic we substitute $$x=y-\frac b{3a}$$ to get $$a {\left[y-\frac b{3a}\right]}^3+b{\left[y-\frac b{3a}\right]}^2+c{\left[y-\frac b{3a}\right]}+d=0$$

Which simplyfies to $$ay^3+y \left[c-\frac b{3a}\right]+\left[d+\frac{2b^3}{27a^2}-\frac{bc}{3a}\right]=0$$

Now we have a depressed cubic so now we can just simply use the shortened Cardano's formula which is, $$x={\left[\frac{-d}2+\sqrt{\frac{d^2}4+ \frac{c^3}{27}}\right]}^\frac13 +{\left[\frac{-d}2-\sqrt{\frac{d^2}4+ \frac{c^3}{27}}\right]}^\frac13$$

Just put up the values and answer will pop out!

P.S. It took a hell lot of formatting to do this. Anyone knows how to do it faster?.

Better way to do it is this. If you write $$t=3x$$ you get $$t^3+7t+8=0$$

so $$t^3+1+7t+7=0$$ $$(t+1)(t^2-t+1)+7(t+1)=0$$

Now it is easy to finish.

Yes it exists. It's called Cardano's Method of Solving Cubics of the form of cubic you posted. This form is called a depressed cubic because it has the $$x^2$$ term missing.

$$27x^3+21x+8=0$$ $$27x^3+1+21x+7=0$$ Factoring,we get $$(3x+1)(9x^2+6x+1)+7(3x+1)=0$$ $$(3x+1)(9x^2+6x+8)=0$$
We get answer as $$x=-\frac13$$