1/2r=15-4.5r

Solution for 1/2r=15-4.5r equation:

1/2r=15-4.5r

We move all terms to the left:

1/2r-(15-4.5r)=0


Domain of the equation: 2r!=0

r!=0/2

r!=0

r∈R


We add all the numbers together, and all the variables

1/2r-(-4.5r+15)=0

We get rid of parentheses

1/2r+4.5r-15=0

We multiply all the terms by the denominator


(4.5r)*2r-15*2r+1=0

We add all the numbers together, and all the variables

(+4.5r)*2r-15*2r+1=0

We multiply parentheses

8r^2-15*2r+1=0

Wy multiply elements

8r^2-30r+1=0

a = 8; b = -30; c = +1;
Δ = b2-4ac
Δ = -302-4·8·1
Δ = 868
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:

$\sqrt{\Delta}=\sqrt{868}=\sqrt{4*217}=\sqrt{4}*\sqrt{217}=2\sqrt{217}$
$r_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-30)-2\sqrt{217}}{2*8}=\frac{30-2\sqrt{217}}{16}
$
$r_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-30)+2\sqrt{217}}{2*8}=\frac{30+2\sqrt{217}}{16}
$