Esercizio 20. 
Data la seguente equazione
![\[(x^2-3)^6+13(x^2-3)^3+40=0\]](https://quisirisolve.com/wp-content/ql-cache/quicklatex.com-4dc139fa10f6e8ab2c4fb09d3dabda05_l3.png "Rendered by QuickLaTeX.com")
nell’incognita reale 
, determinare l’insieme risolutivo.
Svolgimento. Poniamo 
e otteniamo
![\[y^6+13y^3+40=0;\]](https://quisirisolve.com/wp-content/ql-cache/quicklatex.com-cbae7e5ac74418589ec612f1c9bf71e8_l3.png "Rendered by QuickLaTeX.com")
l’equazione ottenuta è un’equazione trinomia, quindi poniamo 
, ottenendo:
![\[t^2+13t+40=0.\]](https://quisirisolve.com/wp-content/ql-cache/quicklatex.com-c98c91ff3fc2598fcd505ab294fe857f_l3.png "Rendered by QuickLaTeX.com")
Per risolvere l’equazione applichiamo la formula risolutiva:
![\[\Delta=(13)^2-4\cdot40=169-160=9,\quad t_{1,2}=\frac{-13\pm\sqrt{9}}{2}=\frac{-13\pm3}{2}= \left\{ \begin{aligned} t_1&=-5\\ t_2&=-8, \end{aligned} \right.\]](https://quisirisolve.com/wp-content/ql-cache/quicklatex.com-6e8213c88c581069ee20de6d56a25a45_l3.png "Rendered by QuickLaTeX.com")
da cui, procedendo a retroso, si ha:
![\[\begin{aligned} y^3&=-5\quad \Leftrightarrow \quad y=-\sqrt[3]{5}\\ y^3&=-8\quad \Leftrightarrow \quad y=-\sqrt[3]{8}=-2, \end{aligned}\]](https://quisirisolve.com/wp-content/ql-cache/quicklatex.com-ad21ce8d5cbae92f0307ca88bb474a28_l3.png "Rendered by QuickLaTeX.com")
quindi
![\[\begin{aligned} x^2-3&=-\sqrt[3]{5}\quad \Leftrightarrow \quad x^2=3-\sqrt[3]{5}\\ x^2-3&=-2\quad \Leftrightarrow \quad x^2=3-2=1. \end{aligned}\]](https://quisirisolve.com/wp-content/ql-cache/quicklatex.com-6f7fd40aa82b57e2e25f4c1384c40c67_l3.png "Rendered by QuickLaTeX.com")
Dato che ![3-\sqrt[3]{5}](https://quisirisolve.com/wp-content/ql-cache/quicklatex.com-4a1eb6e3c0ad59b229978f8ded5e15e3_l3.png "Rendered by QuickLaTeX.com")
e 
sono numeri positivi, si ottengono le soluzioni
![\[\begin{aligned} &x^2=3-\sqrt[3]{5}\quad \Leftrightarrow \quad x=\pm\sqrt{3-\sqrt[3]{5}},\\ &x^2=1\quad \Leftrightarrow \quad x=\pm\sqrt{1}=\pm1. \end{aligned}\]](https://quisirisolve.com/wp-content/ql-cache/quicklatex.com-04e019fba71763b3da4a45ee3190847a_l3.png "Rendered by QuickLaTeX.com")
Si conclude che la soluzione 
dell’equazione è:
![\[\boxcolorato{analisi}{\mathcal{S}=\left\{-\sqrt{3-\sqrt[3]{5}},-1,1,\sqrt{3-\sqrt[3]{5}}\right\}.}\]](https://quisirisolve.com/wp-content/ql-cache/quicklatex.com-5a11698deb0a88ae1a46edc7282301eb_l3.png "Rendered by QuickLaTeX.com")