1/2x-2=4x+5

Solution for 1/2x-2=4x+5 equation:

1/2x-2=4x+5

We move all terms to the left:

1/2x-2-(4x+5)=0


Domain of the equation: 2x!=0

x!=0/2

x!=0

x∈R


We get rid of parentheses

1/2x-4x-5-2=0

We multiply all the terms by the denominator


-4x*2x-5*2x-2*2x+1=0

Wy multiply elements

-8x^2-10x-4x+1=0

We add all the numbers together, and all the variables

-8x^2-14x+1=0

a = -8; b = -14; c = +1;
Δ = b2-4ac
Δ = -142-4·(-8)·1
Δ = 228
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:

$\sqrt{\Delta}=\sqrt{228}=\sqrt{4*57}=\sqrt{4}*\sqrt{57}=2\sqrt{57}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-14)-2\sqrt{57}}{2*-8}=\frac{14-2\sqrt{57}}{-16}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-14)+2\sqrt{57}}{2*-8}=\frac{14+2\sqrt{57}}{-16}
$