3x-5=1/5x+23

Solution for 3x-5=1/5x+23 equation:

3x-5=1/5x+23

We move all terms to the left:

3x-5-(1/5x+23)=0


Domain of the equation: 5x+23)!=0

x∈R


We get rid of parentheses

3x-1/5x-23-5=0

We multiply all the terms by the denominator


3x*5x-23*5x-5*5x-1=0

Wy multiply elements

15x^2-115x-25x-1=0

We add all the numbers together, and all the variables

15x^2-140x-1=0

a = 15; b = -140; c = -1;
Δ = b2-4ac
Δ = -1402-4·15·(-1)
Δ = 19660
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:

$\sqrt{\Delta}=\sqrt{19660}=\sqrt{4*4915}=\sqrt{4}*\sqrt{4915}=2\sqrt{4915}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-140)-2\sqrt{4915}}{2*15}=\frac{140-2\sqrt{4915}}{30}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-140)+2\sqrt{4915}}{2*15}=\frac{140+2\sqrt{4915}}{30}
$