# \[?\] Solve a cubic equation? **URL:** \ **Category:** Questions\ **Tags:** equation-solving\ **Created:** [November 15, 2017, 9:21am UTC](https://community.wolfram.com/t/solve-a-cubic-equation/11527 "2017-11-15T09:21:51Z")\ **Posts on this page:** 7\ **Page:** 1
**Author:** ![guneymetin](https://avatars.discourse-cdn.com/v4/letter/g/b782af/32.png) [@guneymetin](https://community.wolfram.com/u/guneymetin)\ **Post date:** [November 15, 2017, 9:21am UTC](https://community.wolfram.com/t/solve-a-cubic-equation/11527/1 "2017-11-15T09:21:51Z")
Please look at this outcome: ![ucuncuDD_forumSoru1.PNG](https://global.discourse-cdn.com/wolfram/original/2X/f/f8a4362900c9fe3af01439657ffb1c9e6ae401e8.png) As you see, all 3 roots are complex numbers. But this equition has a real root at least 1. This is 0.1736 … (exactly sin 10 degrees). ![ucuncuDD_forumSoru2.PNG](https://global.discourse-cdn.com/wolfram/original/3X/1/1/111836407b46318ab936aa112921b0db1ffb3181.png) Where is my fault? [1]: [http://community.wolfram.com//c/portal/getImageAttachment?filename=ucuncuDD\_forumSoru1.PNG&userId=1222434](http://community.wolfram.com//c/portal/getImageAttachment?filename=ucuncuDD_forumSoru1.PNG&userId=1222434) [2]: [http://community.wolfram.com//c/portal/getImageAttachment?filename=ucuncuDD\_forumSoru2.PNG&userId=1222434](http://community.wolfram.com//c/portal/getImageAttachment?filename=ucuncuDD_forumSoru2.PNG&userId=1222434) ---
**Author:** ![priyanfernando](https://avatars.discourse-cdn.com/v4/letter/p/50afbb/32.png) [@priyanfernando](https://community.wolfram.com/u/priyanfernando)\ **Post date:** [November 15, 2017, 9:48am UTC](https://community.wolfram.com/t/solve-a-cubic-equation/11527/2 "2017-11-15T09:48:18Z")
You can use NSolve: ``` NSolve[(8*t^3 - 6*t + 1 == 0), t] ``` This gives {{t → -0.939693}, {t → 0.173648}, {t → 0.766044}} For exact solutions: ``` ComplexExpand@Solve[(8*t^3 - 6*t + 1 == 0), t] ``` ---
**Author:** ![mariuszi28cg](https://yyz1.discourse-cdn.com/wolfram/user_avatar/community.wolfram.com/mariuszi28cg/32/308_2.png) [@mariuszi28cg](https://community.wolfram.com/u/mariuszi28cg)\ **Post date:** [November 15, 2017, 10:25am UTC](https://community.wolfram.com/t/solve-a-cubic-equation/11527/3 "2017-11-15T10:25:28Z")
Mathematica is a mathematical program that performs computations on complex numbers(complex plain) and therefore we see the imaginary unit **I** everywhere. If You want solution try with **ComplexExpand** : ``` sol = Solve[8*t^3 - 6*t + 1 == 0, t] // ComplexExpand (* {{t -> 1/2 Cos[\[Pi]/9] + 1/2 Sqrt[3] Sin[\[Pi]/9]}, {t -> 1/2 Cos[\[Pi]/9] - 1/2 Sqrt[3] Sin[\[Pi]/9]}, {t -> -Cos[\[Pi]/9]}} *) sol//N (* {{t -> 0.766044}, {t -> 0.173648}, {t -> -0.939693}} *) ``` Yours equation has 3 real roots: ``` Plot[8*t^3 - 6*t + 1, {t, -1, 1}, Epilog -> {Red, PointSize[0.02], Point[{t /. sol[[#]] // N, 0} & /@ {1, 2, 3}]}, AxesLabel -> {t, f[t]}, PlotLabels -> Placed[{"8*t^3-6*t+1"}, Above]] ``` ![6760Bez tytułu.png](https://global.discourse-cdn.com/wolfram/original/2X/c/ccb020b77fae0462ca778f5fe8ec520f50a27152.png) Another example with 1 real and 2 complex roots: ``` sol1 = Solve[t^3 - t + 1 == 0, t] // ComplexExpand (* {{t -> -(2/(3 (9 - Sqrt[69])))^(1/3) - (1/2 (9 - Sqrt[69]))^(1/3)/3^( 2/3)}, {t -> (1/2 (9 - Sqrt[69]))^(1/3)/(2 3^(2/3)) + 1/( 2^(2/3) (3 (9 - Sqrt[69]))^(1/3)) + I (-(3^(1/6)/(2^(2/3) (9 - Sqrt[69])^(1/3))) + (9 - Sqrt[69])^( 1/3)/(2 2^(1/3) 3^(1/6)))}, {t -> (1/2 (9 - Sqrt[69]))^(1/3)/( 2 3^(2/3)) + 1/(2^(2/3) (3 (9 - Sqrt[69]))^(1/3)) + I (3^(1/6)/(2^(2/3) (9 - Sqrt[69])^(1/3)) - (9 - Sqrt[69])^(1/3)/( 2 2^(1/3) 3^(1/6)))}}*) sol1 // N (* {{t -> -1.32472}, {t -> 0.662359 - 0.56228 I}, {t -> 0.662359 + 0.56228 I}} *) Plot[t^3 - t + 1, {t, -2, 2}, Epilog -> {Red, PointSize[0.02], Point[{t /. sol1[[#]] // N, 0} & /@ {1}]}, AxesLabel -> {t, f[t]}, PlotLabels -> Placed[{"t^3-t+1"}, Above]] ``` ![222.png](https://global.discourse-cdn.com/wolfram/original/2X/3/342974dd295306661151bd43a7dd101dfa36bbac.png) ---
**Author:** ![guneymetin](https://avatars.discourse-cdn.com/v4/letter/g/b782af/32.png) [@guneymetin](https://community.wolfram.com/u/guneymetin)\ **Post date:** [November 15, 2017, 10:43am UTC](https://community.wolfram.com/t/solve-a-cubic-equation/11527/4 "2017-11-15T10:43:26Z")
Thanks for your explanations. But what I can’t understand is the behavior of Mathematica. Is this equition has more than 3 roots, 3 of them are real, other 3 are complex? ---
**Author:** ![danl](https://yyz1.discourse-cdn.com/wolfram/user_avatar/community.wolfram.com/danl/32/41267_2.png) [@danl](https://community.wolfram.com/u/danl)\ **Post date:** [November 15, 2017, 4:05pm UTC](https://community.wolfram.com/t/solve-a-cubic-equation/11527/5 "2017-11-15T16:05:26Z")
(1) Please post actual code rather than an image of code. (2) Have a look at the [casus irreducibilus](https://en.wikipedia.org/wiki/Casus_irreducibilis). (3) One can avoid seemingly complex radicals in the result using an option setting to disable the Cardano/Tartaglia cubic form of result. ``` InputForm[Solve[(8*t^3 - 6*t + 1 == 0), t, Cubics->False]] (* Out[2]//InputForm= {{t -> Root[1 - 6*#1 + 8*#1^3 & , 1, 0]}, {t -> Root[1 - 6*#1 + 8*#1^3 & , 2, 0]}, {t -> Root[1 - 6*#1 + 8*#1^3 & , 3, 0]}} *) N[%] (* Out[3]= {{t -> -0.939693}, {t -> 0.173648}, {t -> 0.766044}} *) ``` Alternatively convert the cubic radicals to trigs. ``` InputForm[ComplexExpand[Solve[(8*t^3 - 6*t + 1 == 0), t]]] (* Out[5]//InputForm= {{t -> Cos[Pi/9]/2 + (Sqrt[3]*Sin[Pi/9])/2}, {t -> Cos[Pi/9]/2 - (Sqrt[3]*Sin[Pi/9])/2}, {t -> -Cos[Pi/9]}} *) ``` ---
**Author:** ![guneymetin](https://avatars.discourse-cdn.com/v4/letter/g/b782af/32.png) [@guneymetin](https://community.wolfram.com/u/guneymetin)\ **Post date:** [November 20, 2017, 7:00am UTC](https://community.wolfram.com/t/solve-a-cubic-equation/11527/6 "2017-11-20T07:00:50Z")
Thanks for detailed explanations. ---
**Author:** ![guneymetin](https://avatars.discourse-cdn.com/v4/letter/g/b782af/32.png) [@guneymetin](https://community.wolfram.com/u/guneymetin)\ **Post date:** [November 20, 2017, 7:06am UTC](https://community.wolfram.com/t/solve-a-cubic-equation/11527/7 "2017-11-20T07:06:27Z")
“Casus irreducibilus” explained everything. Thank you for detailed explanations.