Question

Will give Brainliest! Just correct answers with a full explanation. A Simple way to solve x-2%5Ccdot%205%5E%7B2x%7D-10%5E%7Bx%7D%5C%20%20%5Ctextgreater%20%5C%200" id="TexFormula1" title="4^x-2\cdot 5^{2x}-10^{x}\ \textgreater \ 0" alt="4^x-2\cdot 5^{2x}-10^{x}\ \textgreater \ 0" align="absmiddle" class="latex-formula"> The answer is x < -0.756471...
Approximation of $x\ \textless \ \frac{\log(2)}{\log(2)-\log(5)}$ or $x\ \textless \ \log_\frac{5}{2} \left(\frac{1}{2} \right)$
I could solve it factoring, but it took time.

Answer 1

$4^x-2\cdot5^{2x}-10^x>0$

$\implies 2^{2x}-2\cdot5^{2x}-(2^x)(5^x)>0$

let $2^x=a$ and $5^x=b$

So, $a^2-2b^2-ab>0$

divide by $b^2$, ($b^2>0$)

$\implies \left(\frac ab\right)^2-\left(\frac ab\right) -2>0$

this is a quadratic, in $\left(\frac ab\right)$, let it be $x$

So, $x^2-x-2>0$

Can you simplify it now?