(1/2x+1)=3x+4

Solution for (1/2x+1)=3x+4 equation:

(1/2x+1)=3x+4

We move all terms to the left:

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


Domain of the equation: 2x+1)!=0

x∈R


We get rid of parentheses

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

We multiply all the terms by the denominator


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

Wy multiply elements

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

We add all the numbers together, and all the variables

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

a = -6; b = -6; c = +1;
Δ = b2-4ac
Δ = -62-4·(-6)·1
Δ = 60
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:

$\sqrt{\Delta}=\sqrt{60}=\sqrt{4*15}=\sqrt{4}*\sqrt{15}=2\sqrt{15}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-6)-2\sqrt{15}}{2*-6}=\frac{6-2\sqrt{15}}{-12}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-6)+2\sqrt{15}}{2*-6}=\frac{6+2\sqrt{15}}{-12}
$