source: pyromaths/trunk/fuentes/data/ex/lycee/tests/Sd2aRacines.0.answer @ 423

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1\exercice*
2Déterminer les racines des polynômes :\par
3\begin{tabularx}{\linewidth}[t]{XX}
4{$\! \begin{aligned}
5P\,(x) &= 6\,x^{2}+7\,x\\
6&=x\times \left( 6\,x+7\right)
7\end{aligned}$}\par
8\underline{Les racines de $P\,(x)$ sont }\fbox{$0$}\underline{ et }\fbox{$\dfrac{-7}{6}$}
9&
10{$\! \begin{aligned}
11R\,(x) &= 4\,x^{2}-64\\
12&=\left( \sqrt{4}\,x\right) ^{2}-\sqrt{64}^{2}\\
13&=\left( \sqrt{4}\,x+\sqrt{64}\right) \times \left( \sqrt{4}\,x-\sqrt{64}\right) \\
14&=\left( 2\,x+8\right) \times \left( 2\,x-8\right)
15\end{aligned}$}\par
16\underline{Les racines de $R\,(x)$ sont }\fbox{$-4$}\underline{ et }\fbox{$4$}
17\end{tabularx}\par\medskip
18$Q\,(x) = x^{2}+12\,x-9\quad$
19On calcule le discriminant de $Q\,(x)$ avec $a=1$, $b=12$ et $c=-9$ :\par\medskip
20\begin{tabularx}{\linewidth}[t]{XXX}
21{$\! \begin{aligned}
22\Delta &= 12^{2}-4\times 1\times \left( -9\right) \\
23\Delta &= 144-\left( -36\right) \\
24\Delta &= 180\\
25\end{aligned}$}
26&
27{$\! \begin{aligned}
28x_1 &= \dfrac{-12-\sqrt{180}}{2\times 1}\\
29x_1 &= \dfrac{-12-\sqrt{36}\times \sqrt{5}}{2}\\
30x_1 &= \dfrac{\left (  -6-3\,\sqrt{5}\right )  \times \cancel{2}}{1\times \cancel{2}}\\
31x_1 &= -6-3\,\sqrt{5}\\
32\end{aligned}$}
33&
34{$\! \begin{aligned}
35x_2 &= \dfrac{-12+\sqrt{180}}{2\times 1}\\
36x_2 &= \dfrac{-12+\sqrt{36}\times \sqrt{5}}{2}\\
37x_2 &= \dfrac{\left (  -6+3\,\sqrt{5}\right )  \times \cancel{2}}{1\times \cancel{2}}\\
38x_2 &= -6+3\,\sqrt{5}\\
39\end{aligned}$}
40\end{tabularx}\par
41\underline{Les racines de $Q\,(x)$ sont }\fbox{$-6-3\,\sqrt{5}$}\underline{ et }\fbox{$-6+3\,\sqrt{5}$}
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