(1/2x-7)=3x+4

Solution for (1/2x-7)=3x+4 equation:

(1/2x-7)=3x+4

We move all terms to the left:

(1/2x-7)-(3x+4)=0


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

x∈R


We get rid of parentheses

1/2x-3x-7-4=0

We multiply all the terms by the denominator


-3x*2x-7*2x-4*2x+1=0

Wy multiply elements

-6x^2-14x-8x+1=0

We add all the numbers together, and all the variables

-6x^2-22x+1=0

a = -6; b = -22; c = +1;
Δ = b2-4ac
Δ = -222-4·(-6)·1
Δ = 508
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:

$\sqrt{\Delta}=\sqrt{508}=\sqrt{4*127}=\sqrt{4}*\sqrt{127}=2\sqrt{127}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-22)-2\sqrt{127}}{2*-6}=\frac{22-2\sqrt{127}}{-12}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-22)+2\sqrt{127}}{2*-6}=\frac{22+2\sqrt{127}}{-12}
$