x+16=1/x-8

Solution for x+16=1/x-8 equation:

x+16=1/x-8

We move all terms to the left:

x+16-(1/x-8)=0


Domain of the equation: x-8)!=0

x∈R


We get rid of parentheses

x-1/x+8+16=0

We multiply all the terms by the denominator


x*x+8*x+16*x-1=0

We add all the numbers together, and all the variables

24x+x*x-1=0

Wy multiply elements

x^2+24x-1=0

a = 1; b = 24; c = -1;
Δ = b2-4ac
Δ = 242-4·1·(-1)
Δ = 580
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:

$\sqrt{\Delta}=\sqrt{580}=\sqrt{4*145}=\sqrt{4}*\sqrt{145}=2\sqrt{145}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(24)-2\sqrt{145}}{2*1}=\frac{-24-2\sqrt{145}}{2}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(24)+2\sqrt{145}}{2*1}=\frac{-24+2\sqrt{145}}{2}
$