(10x+40/x+4)+9=55x+220

Solution for (10x+40/x+4)+9=55x+220 equation:

(10x+40/x+4)+9=55x+220

We move all terms to the left:

(10x+40/x+4)+9-(55x+220)=0


Domain of the equation: x+4)!=0

x∈R


We get rid of parentheses

10x+40/x-55x+4-220+9=0

We multiply all the terms by the denominator


10x*x-55x*x+4*x-220*x+9*x+40=0

We add all the numbers together, and all the variables

-207x+10x*x-55x*x+40=0

Wy multiply elements

10x^2-55x^2-207x+40=0

We add all the numbers together, and all the variables

-45x^2-207x+40=0

a = -45; b = -207; c = +40;
Δ = b2-4ac
Δ = -2072-4·(-45)·40
Δ = 50049
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:

$\sqrt{\Delta}=\sqrt{50049}=\sqrt{9*5561}=\sqrt{9}*\sqrt{5561}=3\sqrt{5561}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-207)-3\sqrt{5561}}{2*-45}=\frac{207-3\sqrt{5561}}{-90}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-207)+3\sqrt{5561}}{2*-45}=\frac{207+3\sqrt{5561}}{-90}
$