Question

x⁴ — 10x² +9 = 0 essa equação é biquadrada?

Answer 1

✅ Após resolver os cálculos, concluímos que o conjunto solução da referida equação biquadrada é:

  $\Large\displaystyle\text{ \begin{gathered}\boxed{\boxed{\:\:\:\bf S = \{-3,\,-1,\,1,\,3\}\:\:\:}}\end{gathered} }$

Seja a equação biquadrada:

        $\Large\displaystyle\begin{gathered} x^{4} - 10x^{2} + 9 = 0\end{gathered}$

Sabemos que esta equação foi gerada a partir da seguinte função biquadrada:

     $\Large\displaystyle\begin{gathered} f(x) = x^{4} - 10x^{2} + 9\end{gathered}$

Cujos coeficientes são:

                 $\Large\begin{cases} a = 1\\b = -10\\c = 9\end{cases}$

Para calcular as raízes da função biquadrada devemos fazer:

    $\Large\displaystyle\text{ \begin{gathered} x = \pm\sqrt{\frac{-b\pm\sqrt{b^{2} - 4ac}}{2a}}\end{gathered} }$

         $\Large\displaystyle\text{ \begin{gathered} = \pm\sqrt{\frac{-(-10)\pm\sqrt{(-10)^{2} - 4\cdot1\cdot9}}{2\cdot1}}\end{gathered} }$

         $\Large\displaystyle\text{ \begin{gathered} = \pm\sqrt{\frac{10\pm\sqrt{100 - 36}}{2}}\end{gathered} }$

         $\Large\displaystyle\text{ \begin{gathered} = \pm\sqrt{\frac{10\pm\sqrt{64}}{2}}\end{gathered} }$

         $\Large\displaystyle\text{ \begin{gathered} = \pm\sqrt{\frac{10\pm8}{2}}\end{gathered} }$

Encontrando as raízes, temos:

   $\LARGE\begin{cases} x' = -\sqrt{\frac{10 + 8}{2}} = -\sqrt{\frac{18}{2}} = -\sqrt{9} = -3\\x'' = -\sqrt{\frac{10 - 8}{2}} = -\sqrt{\frac{2}{2}} = -\sqrt{1} = -1\\x''' = \sqrt{\frac{10 - 8}{2}} = \sqrt{\frac{2}{2}} = \sqrt{1} = 1\\x'''' = \sqrt{\frac{10 + 8}{2}} = \sqrt{\frac{18}{2}} = \sqrt{9} = 3\end{cases}$

✅ Portanto, o conjunto solução desta função é:

     $\Large\displaystyle\begin{gathered} S = \{-3,\,-1,\,1,\,3\}\end{gathered}$

$\LARGE\displaystyle\text{ \begin{gathered} \underline{\boxed{\boldsymbol{\:\:\:Bons \:estudos!!\:\:\:Boa\: sorte!!\:\:\:}}}\end{gathered} }$

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$\Large\displaystyle\text{ \begin{gathered} \underline{\boxed{\boldsymbol{\:\:\:Observe \:o\:Gr\acute{a}fico!!\:\:\:}}}\end{gathered} }$

Answer 2

Resposta:

$\textsf{Leia abaixo}$

Explicação passo-a-passo:

$\mathsf{ x^4-10x^2+9=0}$

$\mathsf{ a=1\quad b=-10\quad c=9}$

$\mathsf{\Delta=b^2-4\cdot a\cdot c }$

$\mathsf{\Delta=(-10)^2-4\cdot 1\cdot9 }$

$\mathsf{ \Delta=100-36 }$

$\mathsf{\Delta=64 }$

$\mathsf{ x=\pm\sqrt{\dfrac{-b\pm\sqrt\Delta}{2\cdot a}}}$

$\mathsf{x=\pm\sqrt{\dfrac{-(-10)\pm\sqrt{64}}{2\cdot 1} }}$

$\mathsf{ x=\pm\sqrt{\dfrac{10\pm8}{2}}\begin{cases}\sf x_1=+\sqrt{\dfrac{10+8}{2}}=+\sqrt{9}=+3\\\\\sf x_2=-\sqrt{\dfrac{10+8}{2}}=-\sqrt9=-3\\\\\sf x_3=+\sqrt{\dfrac{10-8}{2}}=+\sqrt1=+1\\\\\sf x_4=-\sqrt{\dfrac{10-8}{2}}=-\sqrt1=-1\end{cases}}$

$\boxed{\boxed{ \mathsf{ V=\{-3;~-1;~+1;~+3\}}}}$