10^x*9^x=1000

Solution for 10^x*9^x=1000 equation:

10^x*9^x=1000

We move all terms to the left:

10^x*9^x-(1000)=0

Wy multiply elements

90x^2-1000=0

a = 90; b = 0; c = -1000;
Δ = b2-4ac
Δ = 02-4·90·(-1000)
Δ = 360000
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$\sqrt{\Delta}=\sqrt{360000}=600$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(0)-600}{2*90}=\frac{-600}{180}
=-3+1/3
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(0)+600}{2*90}=\frac{600}{180}
=3+1/3
$