2x+5=1/2x-3

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

2x+5=1/2x-3

We move all terms to the left:

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


Domain of the equation: 2x-3)!=0

x∈R


We get rid of parentheses

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

We multiply all the terms by the denominator


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

Wy multiply elements

4x^2+6x+10x-1=0

We add all the numbers together, and all the variables

4x^2+16x-1=0

a = 4; b = 16; c = -1;
Δ = b2-4ac
Δ = 162-4·4·(-1)
Δ = 272
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:

$\sqrt{\Delta}=\sqrt{272}=\sqrt{16*17}=\sqrt{16}*\sqrt{17}=4\sqrt{17}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(16)-4\sqrt{17}}{2*4}=\frac{-16-4\sqrt{17}}{8}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(16)+4\sqrt{17}}{2*4}=\frac{-16+4\sqrt{17}}{8}
$