10/(x-4)+9x=2

Solution for 10/(x-4)+9x=2 equation:

10/(x-4)+9x=2

We move all terms to the left:

10/(x-4)+9x-(2)=0


Domain of the equation: (x-4)!=0

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

x!=4

x∈R


We add all the numbers together, and all the variables

9x+10/(x-4)-2=0

We multiply all the terms by the denominator


9x*(x-4)-2*(x-4)+10=0

We multiply parentheses

9x^2-36x-2x+8+10=0

We add all the numbers together, and all the variables

9x^2-38x+18=0

a = 9; b = -38; c = +18;
Δ = b2-4ac
Δ = -382-4·9·18
Δ = 796
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:

$\sqrt{\Delta}=\sqrt{796}=\sqrt{4*199}=\sqrt{4}*\sqrt{199}=2\sqrt{199}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-38)-2\sqrt{199}}{2*9}=\frac{38-2\sqrt{199}}{18}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-38)+2\sqrt{199}}{2*9}=\frac{38+2\sqrt{199}}{18}
$