1/2x=4x+11

Solution for 1/2x=4x+11 equation:

1/2x=4x+11

We move all terms to the left:

1/2x-(4x+11)=0


Domain of the equation: 2x!=0

x!=0/2

x!=0

x∈R


We get rid of parentheses

1/2x-4x-11=0

We multiply all the terms by the denominator


-4x*2x-11*2x+1=0

Wy multiply elements

-8x^2-22x+1=0

a = -8; b = -22; c = +1;
Δ = b2-4ac
Δ = -222-4·(-8)·1
Δ = 516
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:

$\sqrt{\Delta}=\sqrt{516}=\sqrt{4*129}=\sqrt{4}*\sqrt{129}=2\sqrt{129}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-22)-2\sqrt{129}}{2*-8}=\frac{22-2\sqrt{129}}{-16}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-22)+2\sqrt{129}}{2*-8}=\frac{22+2\sqrt{129}}{-16}
$