3x+5=1/2x+16

Solution for 3x+5=1/2x+16 equation:

3x+5=1/2x+16

We move all terms to the left:

3x+5-(1/2x+16)=0


Domain of the equation: 2x+16)!=0

x∈R


We get rid of parentheses

3x-1/2x-16+5=0

We multiply all the terms by the denominator


3x*2x-16*2x+5*2x-1=0

Wy multiply elements

6x^2-32x+10x-1=0

We add all the numbers together, and all the variables

6x^2-22x-1=0

a = 6; b = -22; c = -1;
Δ = b2-4ac
Δ = -222-4·6·(-1)
Δ = 508
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:

$\sqrt{\Delta}=\sqrt{508}=\sqrt{4*127}=\sqrt{4}*\sqrt{127}=2\sqrt{127}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-22)-2\sqrt{127}}{2*6}=\frac{22-2\sqrt{127}}{12}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-22)+2\sqrt{127}}{2*6}=\frac{22+2\sqrt{127}}{12}
$