25^x+2=1/625^x

Solution for 25^x+2=1/625^x equation:

25^x+2=1/625^x

We move all terms to the left:

25^x+2-(1/625^x)=0


Domain of the equation: 625^x)!=0

x!=0/1

x!=0

x∈R


We get rid of parentheses

25^x-1/625^x+2=0

We multiply all the terms by the denominator


25^x*625^x+2*625^x-1=0

Wy multiply elements

15625x^2+1250x-1=0

a = 15625; b = 1250; c = -1;
Δ = b2-4ac
Δ = 12502-4·15625·(-1)
Δ = 1625000
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:

$\sqrt{\Delta}=\sqrt{1625000}=\sqrt{62500*26}=\sqrt{62500}*\sqrt{26}=250\sqrt{26}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(1250)-250\sqrt{26}}{2*15625}=\frac{-1250-250\sqrt{26}}{31250}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(1250)+250\sqrt{26}}{2*15625}=\frac{-1250+250\sqrt{26}}{31250}
$