5(r-3)=101/2r-3

Solution for 5(r-3)=101/2r-3 equation:

5(r-3)=101/2r-3

We move all terms to the left:

5(r-3)-(101/2r-3)=0


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

r∈R


We multiply parentheses

5r-(101/2r-3)-15=0

We get rid of parentheses

5r-101/2r+3-15=0

We multiply all the terms by the denominator


5r*2r+3*2r-15*2r-101=0

Wy multiply elements

10r^2+6r-30r-101=0

We add all the numbers together, and all the variables

10r^2-24r-101=0

a = 10; b = -24; c = -101;
Δ = b2-4ac
Δ = -242-4·10·(-101)
Δ = 4616
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:

$\sqrt{\Delta}=\sqrt{4616}=\sqrt{4*1154}=\sqrt{4}*\sqrt{1154}=2\sqrt{1154}$
$r_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-24)-2\sqrt{1154}}{2*10}=\frac{24-2\sqrt{1154}}{20}
$
$r_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-24)+2\sqrt{1154}}{2*10}=\frac{24+2\sqrt{1154}}{20}
$