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

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

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

We move all terms to the left:

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


Domain of the equation: x-2)!=0

x∈R


We get rid of parentheses

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

We multiply all the terms by the denominator


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


We calculate terms in parentheses: +((x+4)-x*x-2), so:

(x+4)-x*x-2

Wy multiply elements

-1x^2+(x+4)-2

We get rid of parentheses

-1x^2+x+4-2

We add all the numbers together, and all the variables

-1x^2+x+2

Back to the equation:

+(-1x^2+x+2)


We add all the numbers together, and all the variables

(-1x^2+x+2)-9x=0

We get rid of parentheses

-1x^2+x-9x+2=0

We add all the numbers together, and all the variables

-1x^2-8x+2=0

a = -1; b = -8; c = +2;
Δ = b2-4ac
Δ = -82-4·(-1)·2
Δ = 72
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:

$\sqrt{\Delta}=\sqrt{72}=\sqrt{36*2}=\sqrt{36}*\sqrt{2}=6\sqrt{2}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-8)-6\sqrt{2}}{2*-1}=\frac{8-6\sqrt{2}}{-2}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-8)+6\sqrt{2}}{2*-1}=\frac{8+6\sqrt{2}}{-2}
$