1/2x+3=4x+5

Solution for 1/2x+3=4x+5 equation:

1/2x+3=4x+5

We move all terms to the left:

1/2x+3-(4x+5)=0


Domain of the equation: 2x!=0

x!=0/2

x!=0

x∈R


We get rid of parentheses

1/2x-4x-5+3=0

We multiply all the terms by the denominator


-4x*2x-5*2x+3*2x+1=0

Wy multiply elements

-8x^2-10x+6x+1=0

We add all the numbers together, and all the variables

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

a = -8; b = -4; c = +1;
Δ = b2-4ac
Δ = -42-4·(-8)·1
Δ = 48
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:

$\sqrt{\Delta}=\sqrt{48}=\sqrt{16*3}=\sqrt{16}*\sqrt{3}=4\sqrt{3}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-4)-4\sqrt{3}}{2*-8}=\frac{4-4\sqrt{3}}{-16}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-4)+4\sqrt{3}}{2*-8}=\frac{4+4\sqrt{3}}{-16}
$