1/2x+3=5x-8

Solution for 1/2x+3=5x-8 equation:

1/2x+3=5x-8

We move all terms to the left:

1/2x+3-(5x-8)=0


Domain of the equation: 2x!=0

x!=0/2

x!=0

x∈R


We get rid of parentheses

1/2x-5x+8+3=0

We multiply all the terms by the denominator


-5x*2x+8*2x+3*2x+1=0

Wy multiply elements

-10x^2+16x+6x+1=0

We add all the numbers together, and all the variables

-10x^2+22x+1=0

a = -10; b = 22; c = +1;
Δ = b2-4ac
Δ = 222-4·(-10)·1
Δ = 524
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:

$\sqrt{\Delta}=\sqrt{524}=\sqrt{4*131}=\sqrt{4}*\sqrt{131}=2\sqrt{131}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(22)-2\sqrt{131}}{2*-10}=\frac{-22-2\sqrt{131}}{-20}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(22)+2\sqrt{131}}{2*-10}=\frac{-22+2\sqrt{131}}{-20}
$