1/2x=4x+14

Solution for 1/2x=4x+14 equation:

1/2x=4x+14

We move all terms to the left:

1/2x-(4x+14)=0


Domain of the equation: 2x!=0

x!=0/2

x!=0

x∈R


We get rid of parentheses

1/2x-4x-14=0

We multiply all the terms by the denominator


-4x*2x-14*2x+1=0

Wy multiply elements

-8x^2-28x+1=0

a = -8; b = -28; c = +1;
Δ = b2-4ac
Δ = -282-4·(-8)·1
Δ = 816
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:

$\sqrt{\Delta}=\sqrt{816}=\sqrt{16*51}=\sqrt{16}*\sqrt{51}=4\sqrt{51}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-28)-4\sqrt{51}}{2*-8}=\frac{28-4\sqrt{51}}{-16}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-28)+4\sqrt{51}}{2*-8}=\frac{28+4\sqrt{51}}{-16}
$