Question

7X^2/3--55X^1/3--8=0

Answer 1

Answer:

$x=\frac{2969\cdot \:2801^{\frac{1}{2}}}{686}-\frac{157135}{686},\:x=-\frac{157135}{686}-\frac{2969\cdot \:2801^{\frac{1}{2}}}{686}$

Step-by-step explanation:

$7x^{\frac{2}{3}}+55x^{\frac{1}{3}}+8=0$

$7\left(x^{\frac{1}{3}}\right)^2+55x^{\frac{1}{3}}+8=0$

Rewrite as if $x^{\frac{1}{3}}=u$:

$7u^2+55u+8=0$

Use quadratic equation:

$\frac{-55\pm \left(55^2-4\cdot \:7\cdot \:8\right)^{\frac{1}{2}}}{2\cdot \:7}$

Solutions for this part:

$u=\frac{-55+2801^{\frac{1}{2}}}{14},\:u=\frac{-55-2801^{\frac{1}{2}}}{14}$

Substitute $u=x^{\frac{1}{3}}$ back in:

$x^{\frac{1}{3}}=\frac{-55+2801^{\frac{1}{2}}}{14}$

$\left(x^{\frac{1}{3}}\right)^3$

$=x^{\frac{1}{3}\cdot \:3}$

$=x$

$\left(\frac{-55+2801^{\frac{1}{2}}}{14}\right)^3$

$=\frac{\left(-55+2801^{\frac{1}{2}}\right)^3}{14^3}$

$=\left(-55\right)^3+3\left(-55\right)^2\cdot \:2801^{\frac{1}{2}}+3\left(-55\right)\left(2801^{\frac{1}{2}}\right)^2+\left(2801^{\frac{1}{2}}\right)^3$

$=11876\cdot \:2801^{\frac{1}{2}}-628540$

$=\frac{11876\cdot \:2801^{\frac{1}{2}}-628540}{14^3}$

$11876\cdot \:2801^{\frac{1}{2}}-628540$

$=4\left(2969\cdot \:2801^{\frac{1}{2}}-157135\right)$

$=\frac{4\left(2969\cdot \:2801^{\frac{1}{2}}-157135\right)}{14^3}$

$=\frac{2^2\left(2969\cdot \:2801^{\frac{1}{2}}-157135\right)}{2^3\cdot \:7^3}$

$=\frac{2969\cdot \:2801^{\frac{1}{2}}-157135}{7^3\cdot \:2^{3-2}}$

$=\frac{2969\cdot \:2801^{\frac{1}{2}}-157135}{7^3\cdot \:2}$

$=\frac{2969\cdot \:2801^{\frac{1}{2}}-157135}{7^3\cdot \:2}$

$=\frac{2969\cdot \:2801^{\frac{1}{2}}}{686}-\frac{157135}{686}$

$x=\frac{2969\cdot \:2801^{\frac{1}{2}}}{686}-\frac{157135}{686}$

Now solve $x^{\frac{1}{3}}=\frac{-55-2801^{\frac{1}{2}}}{14}$:

$x^{\frac{1}{3}}=\frac{-55-2801^{\frac{1}{2}}}{14}$

$\left(x^{\frac{1}{3}}\right)^3=\left(\frac{-55-2801^{\frac{1}{2}}}{14}\right)^3$

$\left(x^{\frac{1}{3}}\right)^3$

$=x^{\frac{1}{3}\cdot \:3}$

$=x$

$\left(\frac{-55-2801^{\frac{1}{2}}}{14}\right)^3$

$=\frac{\left(-55-2801^{\frac{1}{2}}\right)^3}{14^3}$

$\left(-55-2801^{\frac{1}{2}}\right)^3$

$=\left(-55\right)^3-3\left(-55\right)^2\cdot \:2801^{\frac{1}{2}}+3\left(-55\right)\left(2801^{\frac{1}{2}}\right)^2-\left(2801^{\frac{1}{2}}\right)^3$

$=-628540-11876\cdot \:2801^{\frac{1}{2}}$

$=\frac{\left(-628540-11876\cdot \:2801^{\frac{1}{2}}\right)}{14^3}$

$=\frac{-628540-11876\cdot \:2801^{\frac{1}{2}}}{14^3}$

$-628540-11876\cdot \:2801^{\frac{1}{2}}$

$=-4\left(157135+2969\cdot \:2801^{\frac{1}{2}}\right)$

$=-\frac{4\left(157135+2969\cdot \:2801^{\frac{1}{2}}\right)}{14^3}$

$=-\frac{2^2\left(157135+2969\cdot \:2801^{\frac{1}{2}}\right)}{2^3\cdot \:7^3}$

$=-\frac{157135+2969\cdot \:2801^{\frac{1}{2}}}{7^3\cdot \:2^{3-2}}$

$=-\frac{157135+2969\cdot \:2801^{\frac{1}{2}}}{7^3\cdot \:2}$

$=-\frac{157135+2969\cdot \:2801^{\frac{1}{2}}}{686}$

$=-\left(\frac{157135}{686}\right)-\left(\frac{2969\cdot \:2801^{\frac{1}{2}}}{686}\right)$

$=-\frac{157135}{686}-\frac{2969\cdot \:2801^{\frac{1}{2}}}{686}$

$x=-\frac{157135}{686}-\frac{2969\cdot \:2801^{\frac{1}{2}}}{686}$

Solutions:

$x=\frac{2969\cdot \:2801^{\frac{1}{2}}}{686}-\frac{157135}{686},\:x=-\frac{157135}{686}-\frac{2969\cdot \:2801^{\frac{1}{2}}}{686}$