1/2x-3=5x+6

Solution for 1/2x-3=5x+6 equation:

1/2x-3=5x+6

We move all terms to the left:

1/2x-3-(5x+6)=0


Domain of the equation: 2x!=0

x!=0/2

x!=0

x∈R


We get rid of parentheses

1/2x-5x-6-3=0

We multiply all the terms by the denominator


-5x*2x-6*2x-3*2x+1=0

Wy multiply elements

-10x^2-12x-6x+1=0

We add all the numbers together, and all the variables

-10x^2-18x+1=0

a = -10; b = -18; c = +1;
Δ = b2-4ac
Δ = -182-4·(-10)·1
Δ = 364
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:

$\sqrt{\Delta}=\sqrt{364}=\sqrt{4*91}=\sqrt{4}*\sqrt{91}=2\sqrt{91}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-18)-2\sqrt{91}}{2*-10}=\frac{18-2\sqrt{91}}{-20}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-18)+2\sqrt{91}}{2*-10}=\frac{18+2\sqrt{91}}{-20}
$