25^x+1=625/5x

Solution for 25^x+1=625/5x equation:

25^x+1=625/5x

We move all terms to the left:

25^x+1-(625/5x)=0


Domain of the equation: 5x)!=0

x!=0/1

x!=0

x∈R


We add all the numbers together, and all the variables

25^x-(+625/5x)+1=0

We get rid of parentheses

25^x-625/5x+1=0

We multiply all the terms by the denominator


25^x*5x+1*5x-625=0

Wy multiply elements

125x^2+5x-625=0

a = 125; b = 5; c = -625;
Δ = b2-4ac
Δ = 52-4·125·(-625)
Δ = 312525
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:

$\sqrt{\Delta}=\sqrt{312525}=\sqrt{225*1389}=\sqrt{225}*\sqrt{1389}=15\sqrt{1389}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(5)-15\sqrt{1389}}{2*125}=\frac{-5-15\sqrt{1389}}{250}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(5)+15\sqrt{1389}}{2*125}=\frac{-5+15\sqrt{1389}}{250}
$