(4x+8)/x^2-4=5/6

Solution for (4x+8)/x^2-4=5/6 equation:

(4x+8)/x^2-4=5/6

We move all terms to the left:

(4x+8)/x^2-4-(5/6)=0


Domain of the equation: x^2!=0

x^2!=0/

x^2!=√0

x!=0

x∈R


We add all the numbers together, and all the variables

(4x+8)/x^2-4-(+5/6)=0

We get rid of parentheses

(4x+8)/x^2-4-5/6=0

We calculate fractions


(-5x^2)/6x^2+(24x+48)/6x^2-4=0

We multiply all the terms by the denominator


(-5x^2)+(24x+48)-4*6x^2=0

Wy multiply elements

(-5x^2)-24x^2+(24x+48)=0

We get rid of parentheses

-5x^2-24x^2+24x+48=0

We add all the numbers together, and all the variables

-29x^2+24x+48=0

a = -29; b = 24; c = +48;
Δ = b2-4ac
Δ = 242-4·(-29)·48
Δ = 6144
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:

$\sqrt{\Delta}=\sqrt{6144}=\sqrt{1024*6}=\sqrt{1024}*\sqrt{6}=32\sqrt{6}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(24)-32\sqrt{6}}{2*-29}=\frac{-24-32\sqrt{6}}{-58}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(24)+32\sqrt{6}}{2*-29}=\frac{-24+32\sqrt{6}}{-58}
$