Kaila Branch

2022-09-24

no. and nature of roots of ${x}^{\frac{3}{4}\left({\mathrm{log}}_{2}x{\right)}^{2}+{\mathrm{log}}_{2}x-\frac{5}{4}}=\sqrt{2}$
The given equation is
${x}^{\frac{3}{4}\left({\mathrm{log}}_{2}x{\right)}^{2}+{\mathrm{log}}_{2}x-\frac{5}{4}}=\sqrt{2}$
I took ${\mathrm{log}}_{2}x$
and then rewrote the given equation as
${x}^{3{t}^{2}+4t-5}=\sqrt{2}$
But I don't know what to do after this. How will I find the nature and no. of roots?

Bridger Hall

${x}^{\frac{3}{4}\left({\mathrm{log}}_{2}x{\right)}^{2}+{\mathrm{log}}_{2}x-\frac{5}{4}}=\sqrt{2}$
${\mathrm{log}}_{2}{x}^{\frac{3}{4}\left({\mathrm{log}}_{2}x{\right)}^{2}+{\mathrm{log}}_{2}x-\frac{5}{4}}={\mathrm{log}}_{2}\sqrt{2}$
${\mathrm{log}}_{2}x\left(\frac{3}{4}\left({\mathrm{log}}_{2}x{\right)}^{2}+{\mathrm{log}}_{2}x-\frac{5}{4}\right)=\frac{1}{2}$
$t={\mathrm{log}}_{2}x$
$3{t}^{3}+4{t}^{2}-5t-2=0$
${t}_{1}=1.$
Can you finish?

Darius Miles

I'd try to take ${\mathrm{log}}_{2}$ of the whole expression and the solve with respect to $t={\mathrm{log}}_{2}x$