Need help with problem. I think it is a characteristic equation where u let y= something
5 2x-1 + 5x=10
To express the equation in plain text, you could write:
5^(2x - 1) + 5^x = 10
To use $\displaystyle \LaTeX$ wrap the following code:
5^{2x-1}+5^x=10
In TEX tags, which you can generate using the $\displaystyle \Sigma$ button on the toolbar. This will give you:
$\displaystyle 5^{2x-1}+5^x=10$
For this equation, I would use a bit of trial and error. Let's let $\displaystyle x=0$:
$\displaystyle 5^{2(0)-1}+5^0=\frac{1}{5}+1=1.2<10$
So, we see $\displaystyle 0<x$. Next try $\displaystyle x=1$...what do you find?
What we should really do here, and I apologize for being thick, is take the original equation and multiply through by 5 to get:
$\displaystyle 5^{2x}+5\cdot5^x=50$
Write as quadratic in $\displaystyle 5^x$ in standard form:
$\displaystyle \left(5^x\right)^2+5\cdot5^x-50=0$
Factor:
$\displaystyle \left(5^x+10\right)\left(5^x-5\right)=0$
As $\displaystyle 0<5^x$ for all real $\displaystyle x$, we are left with:
$\displaystyle 5^x-5=0$
$\displaystyle 5^x=5^1\implies x=1$