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$\begin{equation} \tag{A} ax^3+bx^2+cx=d \end{equation}$

We can define Delta for quadratic equation to check whether the equation has answer or not....for $f(x)$ which contains powers higher of $2$ for Is there any method to see how many acceptable roots the polynomial contains? Consider equation $(A)$. Can we say whether the equation has real root? How many roots of the function above are acceptable?

Adriano
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ehsan alizad
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1 Answers1

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It happens that equations of the form $ax^3+bx^2+cx+d=0$ and cubic polynomials in general, do have their own so-called discriminant $\Delta$. It is defined as: $$ \Delta=b^2c^2-4ac^3-4b^3d-27a^2d^2+18abcd.\, $$ We have $3$ distinct cases:

  • If $\Delta > 0$, then the equation has three distinct real roots.
  • If $\Delta = 0$, then the equation has a multiple root and all its roots are real.
  • If $\Delta < 0$, then the equation has one real root and two nonreal complex conjugate roots.

The solutions can then be obtained using the cubic formula: $$\begin{align}x &= \sqrt[\displaystyle3\,]{\left(\dfrac{-b^3}{27a^3} + \dfrac{bc}{6a^2} - \dfrac{d}{2a}\right) + \sqrt{\left(\dfrac{-b^3}{27a^3} + \dfrac{bc}{6a^2}-\dfrac{d}{2a}\right)^2 + \left(\dfrac{c}{3a}-\dfrac{b^2}{9a^2}\right)^3}}\\ & + \sqrt[\displaystyle3\,]{\left(\dfrac{-b^3}{27a^3}+\dfrac{bc}{6a^2}-\dfrac{d}{2a}\right) - \sqrt{\left(\dfrac{-b^3}{27a^3}+\dfrac{bc}{6a^2}-\dfrac{d}{2a}\right)^2 + \left(\dfrac{c}{3a}-\dfrac{b^2}{9a^2}\right)^3}} - \dfrac{b}{3a}.\end{align}$$

I hope this helps.
Best wishes, $\mathcal H$akim.

Hakim
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    One should note that the Cubic formula is not generally used for solving cubic equations. There are numerous methods for which one can obtain the roots of a cubic which have been developed over centuries. If you check out the Wikipedia entry, it outlines many of the techniques for finding roots including the derivation of the formula. – Ali Caglayan Apr 27 '14 at 12:45
  • @Alizter I know, all what I wanted to point out is that there is an equivalent of the quadratic formula for cubics. – Hakim Apr 27 '14 at 12:47
  • It was more of a note directed to the OP :) – Ali Caglayan Apr 27 '14 at 12:49
  • @Alizter Thank you for that! But I'm not sure whether the OP will see it since his last apparition on this site returns to August 2013. :-p – Hakim Apr 27 '14 at 12:50