x/(9x+8)+(10x)+(8x+10)=180

Solution for x/(9x+8)+(10x)+(8x+10)=180 equation:

x/(9x+8)+(10x)+(8x+10)=180

We move all terms to the left:

x/(9x+8)+(10x)+(8x+10)-(180)=0


Domain of the equation: (9x+8)!=0

We move all terms containing x to the left, all other terms to the right

9x!=-8

x!=-8/9

x!=-8/9

x∈R


We add all the numbers together, and all the variables

10x+x/(9x+8)+(8x+10)-180=0

We get rid of parentheses

10x+x/(9x+8)+8x+10-180=0

We multiply all the terms by the denominator


10x*(9x+8)+x+8x*(9x+8)+10*(9x+8)-180*(9x+8)=0

We add all the numbers together, and all the variables

x+10x*(9x+8)+8x*(9x+8)+10*(9x+8)-180*(9x+8)=0

We multiply parentheses

90x^2+72x^2+x+80x+64x+90x-1620x+80-1440=0

We add all the numbers together, and all the variables

162x^2-1385x-1360=0

a = 162; b = -1385; c = -1360;
Δ = b2-4ac
Δ = -13852-4·162·(-1360)
Δ = 2799505
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-1385)-\sqrt{2799505}}{2*162}=\frac{1385-\sqrt{2799505}}{324}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-1385)+\sqrt{2799505}}{2*162}=\frac{1385+\sqrt{2799505}}{324}
$