1/2x+5=4x-2

Solution for 1/2x+5=4x-2 equation:

1/2x+5=4x-2

We move all terms to the left:

1/2x+5-(4x-2)=0


Domain of the equation: 2x!=0

x!=0/2

x!=0

x∈R


We get rid of parentheses

1/2x-4x+2+5=0

We multiply all the terms by the denominator


-4x*2x+2*2x+5*2x+1=0

Wy multiply elements

-8x^2+4x+10x+1=0

We add all the numbers together, and all the variables

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

a = -8; b = 14; c = +1;
Δ = b2-4ac
Δ = 142-4·(-8)·1
Δ = 228
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:

$\sqrt{\Delta}=\sqrt{228}=\sqrt{4*57}=\sqrt{4}*\sqrt{57}=2\sqrt{57}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(14)-2\sqrt{57}}{2*-8}=\frac{-14-2\sqrt{57}}{-16}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(14)+2\sqrt{57}}{2*-8}=\frac{-14+2\sqrt{57}}{-16}
$