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Cubic Equation Formula

All cubic equations have either one real root, or three real roots. If the polynomials have the degree three, they are known as cubic polynomials.

The  cubic  equation is of the form,

\[\LARGE ax^{3}+bx^{2}+cx+d=0\]

Depressing the Cubic Equation

Substitute $\large x= y-\frac{b}{3a}$ in the above cubic equation, then we get,

$\large a\left ( y-\frac{b}{3a} \right )^{3}+b\left ( y-\frac{b}{3a} \right )^{2}+c\left ( y-\frac{b}{3a} \right )+d=0$

Simplifying further, we obtain the following depressed cubic equation –

$\large ay^{3}+\left ( c-\frac{b^{2}}{3a} \right )y+\left ( d+\frac{2b^{3}}{27a^{2}} +\frac{bc}{3a}\right )=0$

It must have the term in $x^{3}$ or it would not be cubic ( and so a≠0), but any or all of b, c and d can be zero. For instance:

For instance: $x^{3}-6\times 2+11x-6=0\;or\;4x^{3}+57=0\;or\;x^{3}+9x=0$

Solved Examples

Question 1: Solve $x^{3}-6\times 2+11x-6=0$

This equation can be factorised to give


This equation has three real roots, all different – the solutions are x = 1, x = 2 and x = 3.