-1/2x+6=4x-3

Solution for -1/2x+6=4x-3 equation:

-1/2x+6=4x-3

We move all terms to the left:

-1/2x+6-(4x-3)=0


Domain of the equation: 2x!=0

x!=0/2

x!=0

x∈R


We get rid of parentheses

-1/2x-4x+3+6=0

We multiply all the terms by the denominator


-4x*2x+3*2x+6*2x-1=0

Wy multiply elements

-8x^2+6x+12x-1=0

We add all the numbers together, and all the variables

-8x^2+18x-1=0

a = -8; b = 18; c = -1;
Δ = b2-4ac
Δ = 182-4·(-8)·(-1)
Δ = 292
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:

$\sqrt{\Delta}=\sqrt{292}=\sqrt{4*73}=\sqrt{4}*\sqrt{73}=2\sqrt{73}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(18)-2\sqrt{73}}{2*-8}=\frac{-18-2\sqrt{73}}{-16}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(18)+2\sqrt{73}}{2*-8}=\frac{-18+2\sqrt{73}}{-16}
$