345+x=456+1/x

Solution for 345+x=456+1/x equation:

345+x=456+1/x

We move all terms to the left:

345+x-(456+1/x)=0


Domain of the equation: x)!=0

x!=0/1

x!=0

x∈R


We add all the numbers together, and all the variables

x-(1/x+456)+345=0

We get rid of parentheses

x-1/x-456+345=0

We multiply all the terms by the denominator


x*x-456*x+345*x-1=0

We add all the numbers together, and all the variables

-111x+x*x-1=0

Wy multiply elements

x^2-111x-1=0

a = 1; b = -111; c = -1;
Δ = b2-4ac
Δ = -1112-4·1·(-1)
Δ = 12325
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:

$\sqrt{\Delta}=\sqrt{12325}=\sqrt{25*493}=\sqrt{25}*\sqrt{493}=5\sqrt{493}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-111)-5\sqrt{493}}{2*1}=\frac{111-5\sqrt{493}}{2}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-111)+5\sqrt{493}}{2*1}=\frac{111+5\sqrt{493}}{2}
$