Question

Encontre as soluções reais da equação:
2³√2x-1 = x³ + 1​

Answer 1

Resposta

$\sf \boxed{\sf x=1 }\Rightarrow\boxed{\sf x\approx \:0.61803\dots }\Rightarrow\boxed{\sf \:x\approx \:-1.61803\dots }$

Explicação passo-a-passo:

• Para essa questão temos 2(duas) formas para responder:

1. Resolução tirando potência
2. Resolução pelo Substituto

• Vamos usar os dois:

Resolução tirando potência :

$\sf \large{\sf 2\sqrt[\sf 3]{\sf 2x-1}=x^3+1}\\\\\\\sf$

• Eleve cada lado da equação por 3

$\sf 16x-8=x^9+3x^6+3x^3+1\\\\\\\sf x^9+3x^6+3x^3+1=16x-8\\\\\\\sf x^9+3x^6+3x^3+1+8=16x-8+8\\\\\\\sf x^9+3x^6+3x^3+9=16x\\\\\\\sf x^9+3x^6+3x^3+9-16x=16x-16x\\\\\\\sf x^9+3x^6+3x^3-16x+9=0\\\\\\\sf \left(x-1\right)\left(x^8+x^7+x^6+4x^5+4x^4+4x^3+7x^2+7x-9\right)=0\\\\\\\boxed{\sf Usando\:o\:princ\acute{i}pio\:do\:fator\:zero:\quad \:Se\:ab=0\:entao\:a=0\:ou\:b=0}\\\\\\\sf x-1=0\quad \Rightarrow\quad \:x^8+x^7+x^6+4x^5+4x^4+4x^3+7x^2+7x-9=0$

$\sf \boxed{\sf x=1 }\Rightarrow\boxed{\sf x\approx \:0.61803\dots }\Rightarrow\boxed{\sf \:x\approx \:-1.61803\dots }$

Resolução pelo Substituto

$\sf \sf \large{\sf 2\sqrt[\sf 3]{\sf 2x-1}=x^3+1}\\\\\\\sf$

• Reescreva a equação com $\sf \large{\sf 2\sqrt[\sf 3]{\sf 2x-1}=x^3+1}\sf ~ e ~x=\dfrac{u^3+1}{2}$

$\sf Resolver \Rightarrow2u=\left(\dfrac{u^3+1}{2}\right)^3+1\\\\\\\sf \displaystyle 2u=\frac{u^9}{8}+\frac{3u^6}{8}+\frac{3u^3}{8}+\frac{1}{8}+1\\\\\\\sf \frac{u^9}{8}+\frac{3u^6}{8}+\frac{3u^3}{8}+\frac{1}{8}+1=2u\\\\\\\sf \frac{u^9}{8}+\frac{3u^6}{8}+\frac{3u^3}{8}+\frac{1}{8}+1-2u=2u-2u\\\\\\\sf \frac{1\cdot \:u^9}{8}+\frac{3u^6}{8}+\frac{3u^3}{8}-2u+\frac{1}{8}+1=0$

• Utilize a fórmula de newton e Raphson para Resolver $\sf 0.125u^9+0.375u^6+0.375u^3-2u+0.125=0$

$\sf \displaystyle Aplicar\:a\:divisao\:longa\:Equacao0:\quad \frac{\frac{1\cdot \:u^9}{8}+\frac{3u^6}{8}+\frac{3u^3}{8}-2u+\frac{1}{8}+1}{u-1}=0.125u^8+0.125u^7+0.125u^6+0.5u^5+0.5u^4+0.5u^3+0.875u^2+0.875u-1.125$

$\sf \displaystyle \frac{0.125u^8+0.125u^7+0.125u^6+0.5u^5+0.5u^4+0.5u^3+0.875u^2+0.875u-1.125}{u-0.61803\dots }0.125u^7+0.20225\dots u^6+0.25u^5+0.65450\dots u^4+0.90450\dots u^3+1.05901\dots u^2+1.52950\dots u+1.82028\approx0$

$\sf 0.125u^6+0.25u^4+0.25u^3+0.5u^2+0.25u+1.125\approx \:0\\\\\\\sf u\approx \:1,\:u\approx \:0.61803\dots ,\:u\approx \:-1.61803\dots \\\\\\\sf$

• Substitua  $\sf u=\sf \large{\sf \sqrt[\sf 3]{\sf 2x-1}}\\\\\\\sf$, e solucione x

$\sf \boxed{\begin{array}{lr}\sf \large\:\sqrt[\sf3]{\sf 2x-1}=-1.61803\\\\\\\sf 2x-1=1 \\\\\\\sf x=1\end{array}}\\\\\sf \boxed{\begin{array}{lr}\sf \large\:\sqrt[\sf3]{\sf 2x-1}=0.61803\\\\\\\sf 2x-1=0.23606\dots \\\\\\\sf x=0.61803\dots \end{array}}\\\\\sf \boxed{\begin{array}{lr}\sf \large\:\sqrt[\sf3]{\sf 2x-1}=0.61803\\\\\\\sf 2x-1=-4.23603\dots \\\\\\\sf x=-1.61801\dots \end{array}}$

$\sf \boxed{\sf x=1 }\Rightarrow\boxed{\sf x\approx \:0.61803\dots }\Rightarrow\boxed{\sf \:x\approx \:-1.61803\dots }$