25+r=1/2r-3

Solution for 25+r=1/2r-3 equation:

25+r=1/2r-3

We move all terms to the left:

25+r-(1/2r-3)=0


Domain of the equation: 2r-3)!=0

r∈R


We get rid of parentheses

r-1/2r+3+25=0

We multiply all the terms by the denominator


r*2r+3*2r+25*2r-1=0

Wy multiply elements

2r^2+6r+50r-1=0

We add all the numbers together, and all the variables

2r^2+56r-1=0

a = 2; b = 56; c = -1;
Δ = b2-4ac
Δ = 562-4·2·(-1)
Δ = 3144
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$r_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$r_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{3144}=\sqrt{4*786}=\sqrt{4}*\sqrt{786}=2\sqrt{786}$
$r_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(56)-2\sqrt{786}}{2*2}=\frac{-56-2\sqrt{786}}{4}
$
$r_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(56)+2\sqrt{786}}{2*2}=\frac{-56+2\sqrt{786}}{4}
$