1/4r+4=3r-7

Solution for 1/4r+4=3r-7 equation:

1/4r+4=3r-7

We move all terms to the left:

1/4r+4-(3r-7)=0


Domain of the equation: 4r!=0

r!=0/4

r!=0

r∈R


We get rid of parentheses

1/4r-3r+7+4=0

We multiply all the terms by the denominator


-3r*4r+7*4r+4*4r+1=0

Wy multiply elements

-12r^2+28r+16r+1=0

We add all the numbers together, and all the variables

-12r^2+44r+1=0

a = -12; b = 44; c = +1;
Δ = b2-4ac
Δ = 442-4·(-12)·1
Δ = 1984
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:

$\sqrt{\Delta}=\sqrt{1984}=\sqrt{64*31}=\sqrt{64}*\sqrt{31}=8\sqrt{31}$
$r_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(44)-8\sqrt{31}}{2*-12}=\frac{-44-8\sqrt{31}}{-24}
$
$r_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(44)+8\sqrt{31}}{2*-12}=\frac{-44+8\sqrt{31}}{-24}
$