1/2y+5=y-3

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

1/2y+5=y-3

We move all terms to the left:

1/2y+5-(y-3)=0


Domain of the equation: 2y!=0

y!=0/2

y!=0

y∈R


We get rid of parentheses

1/2y-y+3+5=0

We multiply all the terms by the denominator


-y*2y+3*2y+5*2y+1=0

Wy multiply elements

-2y^2+6y+10y+1=0

We add all the numbers together, and all the variables

-2y^2+16y+1=0

a = -2; b = 16; c = +1;
Δ = b2-4ac
Δ = 162-4·(-2)·1
Δ = 264
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$y_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$y_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{264}=\sqrt{4*66}=\sqrt{4}*\sqrt{66}=2\sqrt{66}$
$y_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(16)-2\sqrt{66}}{2*-2}=\frac{-16-2\sqrt{66}}{-4}
$
$y_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(16)+2\sqrt{66}}{2*-2}=\frac{-16+2\sqrt{66}}{-4}
$