Question

Solve the following equations:
$6x^{\frac{3}{4}}=7x^{\frac{1}{4}}-2x^{-\frac{1}{4}}$.

Answer 1

$6x^{\frac{3}{4}}=7x^{\frac{1}{4}}-2x^{-\frac{1}{4}}6{\left(x^{\frac{1}{4}}\right)}^3=7x^{\frac{1}{4}}-\frac{2}{x^{\frac{1}{4}}}$

Put $x^{\frac{1}{4}}=t$

$6t^3=7t-\frac{2}{t}$

$6t^4=7t^2-2$

$6t^4-7t^2+2=0$

$6t^4-3t^2-4t^2+2=0$

$3t^2(2t^2-1)-2({2t}^2-1)=0$

$(3t^2-2)(2t^2-1)=0$

$t^2=\frac{2}{3},\frac{1}{2}$

$x^{\frac{1}{4}}=t$

$\Rightarrow x=t^4$

$x={\left(\frac{2}{3}\right)}^2,{\left(\frac{1}{2}\right)}^2$

$x=\frac{4}{9},\frac{1}{4}$