x*10^x=5

Solution for x*10^x=5 equation:

x*10^x=5

We move all terms to the left:

x*10^x-(5)=0

Wy multiply elements

10x^2-5=0

a = 10; b = 0; c = -5;
Δ = b2-4ac
Δ = 02-4·10·(-5)
Δ = 200
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{200}=\sqrt{100*2}=\sqrt{100}*\sqrt{2}=10\sqrt{2}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(0)-10\sqrt{2}}{2*10}=\frac{0-10\sqrt{2}}{20}
=-\frac{10\sqrt{2}}{20}
=-\frac{\sqrt{2}}{2}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(0)+10\sqrt{2}}{2*10}=\frac{0+10\sqrt{2}}{20}
=\frac{10\sqrt{2}}{20}
=\frac{\sqrt{2}}{2}
$