4r+9/4r+14=5r-3/4r=1-3

Solution for 4r+9/4r+14=5r-3/4r=1-3 equation:

4r+9/4r+14=5r-3/4r=1-3

We move all terms to the left:

4r+9/4r+14-(5r-3/4r)=0


Domain of the equation: 4r!=0

r!=0/4

r!=0

r∈R



Domain of the equation: 4r)!=0

r!=0/1

r!=0

r∈R


We add all the numbers together, and all the variables

4r+9/4r-(+5r-3/4r)+14=0

We get rid of parentheses

4r+9/4r-5r+3/4r+14=0

We multiply all the terms by the denominator


4r*4r-5r*4r+14*4r+9+3=0

We add all the numbers together, and all the variables

4r*4r-5r*4r+14*4r+12=0

Wy multiply elements

16r^2-20r^2+56r+12=0

We add all the numbers together, and all the variables

-4r^2+56r+12=0

a = -4; b = 56; c = +12;
Δ = b2-4ac
Δ = 562-4·(-4)·12
Δ = 3328
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:

$\sqrt{\Delta}=\sqrt{3328}=\sqrt{256*13}=\sqrt{256}*\sqrt{13}=16\sqrt{13}$
$r_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(56)-16\sqrt{13}}{2*-4}=\frac{-56-16\sqrt{13}}{-8}
$
$r_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(56)+16\sqrt{13}}{2*-4}=\frac{-56+16\sqrt{13}}{-8}
$