1/2x+3=x-7

Solution for 1/2x+3=x-7 equation:

1/2x+3=x-7

We move all terms to the left:

1/2x+3-(x-7)=0


Domain of the equation: 2x!=0

x!=0/2

x!=0

x∈R


We get rid of parentheses

1/2x-x+7+3=0

We multiply all the terms by the denominator


-x*2x+7*2x+3*2x+1=0

Wy multiply elements

-2x^2+14x+6x+1=0

We add all the numbers together, and all the variables

-2x^2+20x+1=0

a = -2; b = 20; c = +1;
Δ = b2-4ac
Δ = 202-4·(-2)·1
Δ = 408
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:

$\sqrt{\Delta}=\sqrt{408}=\sqrt{4*102}=\sqrt{4}*\sqrt{102}=2\sqrt{102}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(20)-2\sqrt{102}}{2*-2}=\frac{-20-2\sqrt{102}}{-4}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(20)+2\sqrt{102}}{2*-2}=\frac{-20+2\sqrt{102}}{-4}
$