x=84+1/8x

Solution for x=84+1/8x equation:

x=84+1/8x

We move all terms to the left:

x-(84+1/8x)=0


Domain of the equation: 8x)!=0

x!=0/1

x!=0

x∈R


We add all the numbers together, and all the variables

x-(1/8x+84)=0

We get rid of parentheses

x-1/8x-84=0

We multiply all the terms by the denominator


x*8x-84*8x-1=0

Wy multiply elements

8x^2-672x-1=0

a = 8; b = -672; c = -1;
Δ = b2-4ac
Δ = -6722-4·8·(-1)
Δ = 451616
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:

$\sqrt{\Delta}=\sqrt{451616}=\sqrt{16*28226}=\sqrt{16}*\sqrt{28226}=4\sqrt{28226}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-672)-4\sqrt{28226}}{2*8}=\frac{672-4\sqrt{28226}}{16}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-672)+4\sqrt{28226}}{2*8}=\frac{672+4\sqrt{28226}}{16}
$