1/2x-5=x+6

Solution for 1/2x-5=x+6 equation:

1/2x-5=x+6

We move all terms to the left:

1/2x-5-(x+6)=0


Domain of the equation: 2x!=0

x!=0/2

x!=0

x∈R


We get rid of parentheses

1/2x-x-6-5=0

We multiply all the terms by the denominator


-x*2x-6*2x-5*2x+1=0

Wy multiply elements

-2x^2-12x-10x+1=0

We add all the numbers together, and all the variables

-2x^2-22x+1=0

a = -2; b = -22; c = +1;
Δ = b2-4ac
Δ = -222-4·(-2)·1
Δ = 492
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:

$\sqrt{\Delta}=\sqrt{492}=\sqrt{4*123}=\sqrt{4}*\sqrt{123}=2\sqrt{123}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-22)-2\sqrt{123}}{2*-2}=\frac{22-2\sqrt{123}}{-4}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-22)+2\sqrt{123}}{2*-2}=\frac{22+2\sqrt{123}}{-4}
$