6+x=7/2x-2

Solution for 6+x=7/2x-2 equation:

6+x=7/2x-2

We move all terms to the left:

6+x-(7/2x-2)=0


Domain of the equation: 2x-2)!=0

x∈R


We get rid of parentheses

x-7/2x+2+6=0

We multiply all the terms by the denominator


x*2x+2*2x+6*2x-7=0

Wy multiply elements

2x^2+4x+12x-7=0

We add all the numbers together, and all the variables

2x^2+16x-7=0

a = 2; b = 16; c = -7;
Δ = b2-4ac
Δ = 162-4·2·(-7)
Δ = 312
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:

$\sqrt{\Delta}=\sqrt{312}=\sqrt{4*78}=\sqrt{4}*\sqrt{78}=2\sqrt{78}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(16)-2\sqrt{78}}{2*2}=\frac{-16-2\sqrt{78}}{4}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(16)+2\sqrt{78}}{2*2}=\frac{-16+2\sqrt{78}}{4}
$