1/2x+27=5x-3

Solution for 1/2x+27=5x-3 equation:

1/2x+27=5x-3

We move all terms to the left:

1/2x+27-(5x-3)=0


Domain of the equation: 2x!=0

x!=0/2

x!=0

x∈R


We get rid of parentheses

1/2x-5x+3+27=0

We multiply all the terms by the denominator


-5x*2x+3*2x+27*2x+1=0

Wy multiply elements

-10x^2+6x+54x+1=0

We add all the numbers together, and all the variables

-10x^2+60x+1=0

a = -10; b = 60; c = +1;
Δ = b2-4ac
Δ = 602-4·(-10)·1
Δ = 3640
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:

$\sqrt{\Delta}=\sqrt{3640}=\sqrt{4*910}=\sqrt{4}*\sqrt{910}=2\sqrt{910}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(60)-2\sqrt{910}}{2*-10}=\frac{-60-2\sqrt{910}}{-20}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(60)+2\sqrt{910}}{2*-10}=\frac{-60+2\sqrt{910}}{-20}
$