1/2x+8=3x+16

Solution for 1/2x+8=3x+16 equation:

1/2x+8=3x+16

We move all terms to the left:

1/2x+8-(3x+16)=0


Domain of the equation: 2x!=0

x!=0/2

x!=0

x∈R


We get rid of parentheses

1/2x-3x-16+8=0

We multiply all the terms by the denominator


-3x*2x-16*2x+8*2x+1=0

Wy multiply elements

-6x^2-32x+16x+1=0

We add all the numbers together, and all the variables

-6x^2-16x+1=0

a = -6; b = -16; c = +1;
Δ = b2-4ac
Δ = -162-4·(-6)·1
Δ = 280
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:

$\sqrt{\Delta}=\sqrt{280}=\sqrt{4*70}=\sqrt{4}*\sqrt{70}=2\sqrt{70}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-16)-2\sqrt{70}}{2*-6}=\frac{16-2\sqrt{70}}{-12}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-16)+2\sqrt{70}}{2*-6}=\frac{16+2\sqrt{70}}{-12}
$