3x-5=-1/2x+7

Solution for 3x-5=-1/2x+7 equation:

3x-5=-1/2x+7

We move all terms to the left:

3x-5-(-1/2x+7)=0


Domain of the equation: 2x+7)!=0

x∈R


We get rid of parentheses

3x+1/2x-7-5=0

We multiply all the terms by the denominator


3x*2x-7*2x-5*2x+1=0

Wy multiply elements

6x^2-14x-10x+1=0

We add all the numbers together, and all the variables

6x^2-24x+1=0

a = 6; b = -24; c = +1;
Δ = b2-4ac
Δ = -242-4·6·1
Δ = 552
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:

$\sqrt{\Delta}=\sqrt{552}=\sqrt{4*138}=\sqrt{4}*\sqrt{138}=2\sqrt{138}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-24)-2\sqrt{138}}{2*6}=\frac{24-2\sqrt{138}}{12}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-24)+2\sqrt{138}}{2*6}=\frac{24+2\sqrt{138}}{12}
$