Respuesta :
Answer:
[tex]x=\frac{2969\cdot \:2801^{\frac{1}{2}}}{686}-\frac{157135}{686},\:x=-\frac{157135}{686}-\frac{2969\cdot \:2801^{\frac{1}{2}}}{686}[/tex]
Step-by-step explanation:
[tex]7x^{\frac{2}{3}}+55x^{\frac{1}{3}}+8=0[/tex]
[tex]7\left(x^{\frac{1}{3}}\right)^2+55x^{\frac{1}{3}}+8=0[/tex]
Rewrite as if [tex]x^{\frac{1}{3}}=u[/tex]:
[tex]7u^2+55u+8=0[/tex]
Use quadratic equation:
[tex]\frac{-55\pm \left(55^2-4\cdot \:7\cdot \:8\right)^{\frac{1}{2}}}{2\cdot \:7}[/tex]
Solutions for this part:
[tex]u=\frac{-55+2801^{\frac{1}{2}}}{14},\:u=\frac{-55-2801^{\frac{1}{2}}}{14}[/tex]
Substitute [tex]u=x^{\frac{1}{3}}[/tex] back in:
[tex]x^{\frac{1}{3}}=\frac{-55+2801^{\frac{1}{2}}}{14}[/tex]
[tex]\left(x^{\frac{1}{3}}\right)^3[/tex]
[tex]=x^{\frac{1}{3}\cdot \:3}[/tex]
[tex]=x[/tex]
[tex]\left(\frac{-55+2801^{\frac{1}{2}}}{14}\right)^3[/tex]
[tex]=\frac{\left(-55+2801^{\frac{1}{2}}\right)^3}{14^3}[/tex]
[tex]=\left(-55\right)^3+3\left(-55\right)^2\cdot \:2801^{\frac{1}{2}}+3\left(-55\right)\left(2801^{\frac{1}{2}}\right)^2+\left(2801^{\frac{1}{2}}\right)^3[/tex]
[tex]=11876\cdot \:2801^{\frac{1}{2}}-628540[/tex]
[tex]=\frac{11876\cdot \:2801^{\frac{1}{2}}-628540}{14^3}[/tex]
[tex]11876\cdot \:2801^{\frac{1}{2}}-628540\\[/tex]
[tex]=4\left(2969\cdot \:2801^{\frac{1}{2}}-157135\right)[/tex]
[tex]=\frac{4\left(2969\cdot \:2801^{\frac{1}{2}}-157135\right)}{14^3}[/tex]
[tex]=\frac{2^2\left(2969\cdot \:2801^{\frac{1}{2}}-157135\right)}{2^3\cdot \:7^3}[/tex]
[tex]=\frac{2969\cdot \:2801^{\frac{1}{2}}-157135}{7^3\cdot \:2^{3-2}}[/tex]
[tex]=\frac{2969\cdot \:2801^{\frac{1}{2}}-157135}{7^3\cdot \:2}[/tex]
[tex]=\frac{2969\cdot \:2801^{\frac{1}{2}}-157135}{7^3\cdot \:2}[/tex]
[tex]=\frac{2969\cdot \:2801^{\frac{1}{2}}}{686}-\frac{157135}{686}[/tex]
[tex]x=\frac{2969\cdot \:2801^{\frac{1}{2}}}{686}-\frac{157135}{686}[/tex]
Now solve [tex]x^{\frac{1}{3}}=\frac{-55-2801^{\frac{1}{2}}}{14}[/tex]:
[tex]x^{\frac{1}{3}}=\frac{-55-2801^{\frac{1}{2}}}{14}[/tex]
[tex]\left(x^{\frac{1}{3}}\right)^3=\left(\frac{-55-2801^{\frac{1}{2}}}{14}\right)^3[/tex]
[tex]\left(x^{\frac{1}{3}}\right)^3[/tex]
[tex]=x^{\frac{1}{3}\cdot \:3}[/tex]
[tex]=x[/tex]
[tex]\left(\frac{-55-2801^{\frac{1}{2}}}{14}\right)^3[/tex]
[tex]=\frac{\left(-55-2801^{\frac{1}{2}}\right)^3}{14^3}[/tex]
[tex]\left(-55-2801^{\frac{1}{2}}\right)^3[/tex]
[tex]=\left(-55\right)^3-3\left(-55\right)^2\cdot \:2801^{\frac{1}{2}}+3\left(-55\right)\left(2801^{\frac{1}{2}}\right)^2-\left(2801^{\frac{1}{2}}\right)^3[/tex]
[tex]=-628540-11876\cdot \:2801^{\frac{1}{2}}[/tex]
[tex]=\frac{\left(-628540-11876\cdot \:2801^{\frac{1}{2}}\right)}{14^3}[/tex]
[tex]=\frac{-628540-11876\cdot \:2801^{\frac{1}{2}}}{14^3}[/tex]
[tex]-628540-11876\cdot \:2801^{\frac{1}{2}}[/tex]
[tex]=-4\left(157135+2969\cdot \:2801^{\frac{1}{2}}\right)[/tex]
[tex]=-\frac{4\left(157135+2969\cdot \:2801^{\frac{1}{2}}\right)}{14^3}[/tex]
[tex]=-\frac{2^2\left(157135+2969\cdot \:2801^{\frac{1}{2}}\right)}{2^3\cdot \:7^3}[/tex]
[tex]=-\frac{157135+2969\cdot \:2801^{\frac{1}{2}}}{7^3\cdot \:2^{3-2}}[/tex]
[tex]=-\frac{157135+2969\cdot \:2801^{\frac{1}{2}}}{7^3\cdot \:2}[/tex]
[tex]=-\frac{157135+2969\cdot \:2801^{\frac{1}{2}}}{686}[/tex]
[tex]=-\left(\frac{157135}{686}\right)-\left(\frac{2969\cdot \:2801^{\frac{1}{2}}}{686}\right)[/tex]
[tex]=-\frac{157135}{686}-\frac{2969\cdot \:2801^{\frac{1}{2}}}{686}[/tex]
[tex]x=-\frac{157135}{686}-\frac{2969\cdot \:2801^{\frac{1}{2}}}{686}[/tex]
Solutions:
[tex]x=\frac{2969\cdot \:2801^{\frac{1}{2}}}{686}-\frac{157135}{686},\:x=-\frac{157135}{686}-\frac{2969\cdot \:2801^{\frac{1}{2}}}{686}[/tex]
