1/2x+6=x-5

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



1/2x+6=x-5
We move all terms to the left:
1/2x+6-(x-5)=0
Domain of the equation: 2x!=0
x!=0/2
x!=0
x∈R
We get rid of parentheses
1/2x-x+5+6=0
We multiply all the terms by the denominator
-x*2x+5*2x+6*2x+1=0
Wy multiply elements
-2x^2+10x+12x+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} $

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