1/2x-3=7x

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

1/2x-3=7x

We move all terms to the left:

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


Domain of the equation: 2x!=0

x!=0/2

x!=0

x∈R


We add all the numbers together, and all the variables

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

We multiply all the terms by the denominator


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

Wy multiply elements

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

a = -14; b = -6; c = +1;
Δ = b2-4ac
Δ = -62-4·(-14)·1
Δ = 92
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:

$\sqrt{\Delta}=\sqrt{92}=\sqrt{4*23}=\sqrt{4}*\sqrt{23}=2\sqrt{23}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-6)-2\sqrt{23}}{2*-14}=\frac{6-2\sqrt{23}}{-28}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-6)+2\sqrt{23}}{2*-14}=\frac{6+2\sqrt{23}}{-28}
$