x-1=1/2x+5

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

x-1=1/2x+5

We move all terms to the left:

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


Domain of the equation: 2x+5)!=0

x∈R


We get rid of parentheses

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

We multiply all the terms by the denominator


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

Wy multiply elements

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

We add all the numbers together, and all the variables

2x^2-12x-1=0

a = 2; b = -12; c = -1;
Δ = b2-4ac
Δ = -122-4·2·(-1)
Δ = 152
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:

$\sqrt{\Delta}=\sqrt{152}=\sqrt{4*38}=\sqrt{4}*\sqrt{38}=2\sqrt{38}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-12)-2\sqrt{38}}{2*2}=\frac{12-2\sqrt{38}}{4}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-12)+2\sqrt{38}}{2*2}=\frac{12+2\sqrt{38}}{4}
$