7-(5/2)(8r-6)+2r=32

Solution for 7-(5/2)(8r-6)+2r=32 equation:

7-(5/2)(8r-6)+2r=32

We move all terms to the left:

7-(5/2)(8r-6)+2r-(32)=0


Domain of the equation: 2)(8r-6)!=0

r∈R


We add all the numbers together, and all the variables

-(+5/2)(8r-6)+2r+7-32=0

We add all the numbers together, and all the variables

2r-(+5/2)(8r-6)-25=0

We multiply parentheses ..

-(+40r^2+5/2*-6)+2r-25=0

We multiply all the terms by the denominator


-(+40r^2+5+2r*2*-6)-25*2*-6)=0

We add all the numbers together, and all the variables

-(+40r^2+5+2r*2*-6)=0

We get rid of parentheses

-40r^2-2r*2*-5+6=0

We add all the numbers together, and all the variables

-40r^2-2r*2*+1=0

Wy multiply elements

-40r^2-4r^2+1=0

We add all the numbers together, and all the variables

-44r^2+1=0

a = -44; b = 0; c = +1;
Δ = b2-4ac
Δ = 02-4·(-44)·1
Δ = 176
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:

$\sqrt{\Delta}=\sqrt{176}=\sqrt{16*11}=\sqrt{16}*\sqrt{11}=4\sqrt{11}$
$r_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(0)-4\sqrt{11}}{2*-44}=\frac{0-4\sqrt{11}}{-88}
=-\frac{4\sqrt{11}}{-88}
=-\frac{\sqrt{11}}{-22}
$
$r_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(0)+4\sqrt{11}}{2*-44}=\frac{0+4\sqrt{11}}{-88}
=\frac{4\sqrt{11}}{-88}
=\frac{\sqrt{11}}{-22}
$