345+x=456+1/x

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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} $

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