4r+9/4r=5r+1-3

Solution for 4r+9/4r=5r+1-3 equation:

4r+9/4r=5r+1-3

We move all terms to the left:

4r+9/4r-(5r+1-3)=0


Domain of the equation: 4r!=0

r!=0/4

r!=0

r∈R


We add all the numbers together, and all the variables

4r+9/4r-(5r-2)=0

We get rid of parentheses

4r+9/4r-5r+2=0

We multiply all the terms by the denominator


4r*4r-5r*4r+2*4r+9=0

Wy multiply elements

16r^2-20r^2+8r+9=0

We add all the numbers together, and all the variables

-4r^2+8r+9=0

a = -4; b = 8; c = +9;
Δ = b2-4ac
Δ = 82-4·(-4)·9
Δ = 208
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:

$\sqrt{\Delta}=\sqrt{208}=\sqrt{16*13}=\sqrt{16}*\sqrt{13}=4\sqrt{13}$
$r_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(8)-4\sqrt{13}}{2*-4}=\frac{-8-4\sqrt{13}}{-8}
$
$r_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(8)+4\sqrt{13}}{2*-4}=\frac{-8+4\sqrt{13}}{-8}
$