5^2x-2+10x/25=5^x+3+125*2^x

Solution for 5^2x-2+10x/25=5^x+3+125*2^x equation:

5^2x-2+10x/25=5^x+3+125*2^x

We move all terms to the left:

5^2x-2+10x/25-(5^x+3+125*2^x)=0

We get rid of parentheses

5^2x+10x/25-5^x-125*2^x-3-2=0

We multiply all the terms by the denominator


5^2x*25+10x-5^x*25-(125*2^x)*25-3*25-2*25=0

We add all the numbers together, and all the variables

10x+5^2x*25-5^x*25-(125*2^x)*25-125=0

We multiply parentheses

10x+5^2x*25-5^x*25-6250x-125=0

Wy multiply elements

125x^2+10x-125x-6250x-125=0

We add all the numbers together, and all the variables

125x^2-6365x-125=0

a = 125; b = -6365; c = -125;
Δ = b2-4ac
Δ = -63652-4·125·(-125)
Δ = 40575725
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:

$\sqrt{\Delta}=\sqrt{40575725}=\sqrt{25*1623029}=\sqrt{25}*\sqrt{1623029}=5\sqrt{1623029}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-6365)-5\sqrt{1623029}}{2*125}=\frac{6365-5\sqrt{1623029}}{250}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-6365)+5\sqrt{1623029}}{2*125}=\frac{6365+5\sqrt{1623029}}{250}
$