1/2x+4=-2x-1

Solution for 1/2x+4=-2x-1 equation:

1/2x+4=-2x-1

We move all terms to the left:

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


Domain of the equation: 2x!=0

x!=0/2

x!=0

x∈R


We get rid of parentheses

1/2x+2x+1+4=0

We multiply all the terms by the denominator


2x*2x+1*2x+4*2x+1=0

Wy multiply elements

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

We add all the numbers together, and all the variables

4x^2+10x+1=0

a = 4; b = 10; c = +1;
Δ = b2-4ac
Δ = 102-4·4·1
Δ = 84
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:

$\sqrt{\Delta}=\sqrt{84}=\sqrt{4*21}=\sqrt{4}*\sqrt{21}=2\sqrt{21}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(10)-2\sqrt{21}}{2*4}=\frac{-10-2\sqrt{21}}{8}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(10)+2\sqrt{21}}{2*4}=\frac{-10+2\sqrt{21}}{8}
$