1/2x-3=5x+10

Solution for 1/2x-3=5x+10 equation:

1/2x-3=5x+10

We move all terms to the left:

1/2x-3-(5x+10)=0


Domain of the equation: 2x!=0

x!=0/2

x!=0

x∈R


We get rid of parentheses

1/2x-5x-10-3=0

We multiply all the terms by the denominator


-5x*2x-10*2x-3*2x+1=0

Wy multiply elements

-10x^2-20x-6x+1=0

We add all the numbers together, and all the variables

-10x^2-26x+1=0

a = -10; b = -26; c = +1;
Δ = b2-4ac
Δ = -262-4·(-10)·1
Δ = 716
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:

$\sqrt{\Delta}=\sqrt{716}=\sqrt{4*179}=\sqrt{4}*\sqrt{179}=2\sqrt{179}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-26)-2\sqrt{179}}{2*-10}=\frac{26-2\sqrt{179}}{-20}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-26)+2\sqrt{179}}{2*-10}=\frac{26+2\sqrt{179}}{-20}
$