1/x-4=6-x

Solution for 1/x-4=6-x equation:

1/x-4=6-x

We move all terms to the left:

1/x-4-(6-x)=0


Domain of the equation: x!=0

x∈R


We add all the numbers together, and all the variables

1/x-(-1x+6)-4=0

We get rid of parentheses

1/x+1x-6-4=0

We multiply all the terms by the denominator


1x*x-6*x-4*x+1=0

We add all the numbers together, and all the variables

-10x+1x*x+1=0

Wy multiply elements

x^2-10x+1=0

a = 1; b = -10; c = +1;
Δ = b2-4ac
Δ = -102-4·1·1
Δ = 96
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:

$\sqrt{\Delta}=\sqrt{96}=\sqrt{16*6}=\sqrt{16}*\sqrt{6}=4\sqrt{6}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-10)-4\sqrt{6}}{2*1}=\frac{10-4\sqrt{6}}{2}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-10)+4\sqrt{6}}{2*1}=\frac{10+4\sqrt{6}}{2}
$