1/2x=5x+12

Solution for 1/2x=5x+12 equation:

1/2x=5x+12

We move all terms to the left:

1/2x-(5x+12)=0


Domain of the equation: 2x!=0

x!=0/2

x!=0

x∈R


We get rid of parentheses

1/2x-5x-12=0

We multiply all the terms by the denominator


-5x*2x-12*2x+1=0

Wy multiply elements

-10x^2-24x+1=0

a = -10; b = -24; c = +1;
Δ = b2-4ac
Δ = -242-4·(-10)·1
Δ = 616
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:

$\sqrt{\Delta}=\sqrt{616}=\sqrt{4*154}=\sqrt{4}*\sqrt{154}=2\sqrt{154}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-24)-2\sqrt{154}}{2*-10}=\frac{24-2\sqrt{154}}{-20}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-24)+2\sqrt{154}}{2*-10}=\frac{24+2\sqrt{154}}{-20}
$