13+6/5x=23-9x

Solution for 13+6/5x=23-9x equation:

13+6/5x=23-9x

We move all terms to the left:

13+6/5x-(23-9x)=0


Domain of the equation: 5x!=0

x!=0/5

x!=0

x∈R


We add all the numbers together, and all the variables

6/5x-(-9x+23)+13=0

We get rid of parentheses

6/5x+9x-23+13=0

We multiply all the terms by the denominator


9x*5x-23*5x+13*5x+6=0

Wy multiply elements

45x^2-115x+65x+6=0

We add all the numbers together, and all the variables

45x^2-50x+6=0

a = 45; b = -50; c = +6;
Δ = b2-4ac
Δ = -502-4·45·6
Δ = 1420
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:

$\sqrt{\Delta}=\sqrt{1420}=\sqrt{4*355}=\sqrt{4}*\sqrt{355}=2\sqrt{355}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-50)-2\sqrt{355}}{2*45}=\frac{50-2\sqrt{355}}{90}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-50)+2\sqrt{355}}{2*45}=\frac{50+2\sqrt{355}}{90}
$