5/x-3=9-x

Solution for 5/x-3=9-x equation:

5/x-3=9-x

We move all terms to the left:

5/x-3-(9-x)=0


Domain of the equation: x!=0

x∈R


We add all the numbers together, and all the variables

5/x-(-1x+9)-3=0

We get rid of parentheses

5/x+1x-9-3=0

We multiply all the terms by the denominator


1x*x-9*x-3*x+5=0

We add all the numbers together, and all the variables

-12x+1x*x+5=0

Wy multiply elements

x^2-12x+5=0

a = 1; b = -12; c = +5;
Δ = b2-4ac
Δ = -122-4·1·5
Δ = 124
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:

$\sqrt{\Delta}=\sqrt{124}=\sqrt{4*31}=\sqrt{4}*\sqrt{31}=2\sqrt{31}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-12)-2\sqrt{31}}{2*1}=\frac{12-2\sqrt{31}}{2}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-12)+2\sqrt{31}}{2*1}=\frac{12+2\sqrt{31}}{2}
$