((6r^2-19r-7)/(2r-7))=4r-3

Solution for ((6r^2-19r-7)/(2r-7))=4r-3 equation:

((6r^2-19r-7)/(2r-7))=4r-3

We move all terms to the left:

((6r^2-19r-7)/(2r-7))-(4r-3)=0


Domain of the equation: (2r-7))!=0

r∈R


We get rid of parentheses

((6r^2-19r-7)/(2r-7))-4r+3=0

We multiply all the terms by the denominator


((6r^2-19r-7)-4r*(2r-7))+3*(2r-7))=0


We calculate terms in parentheses: +((6r^2-19r-7)-4r*(2r-7)), so:

(6r^2-19r-7)-4r*(2r-7)

We multiply parentheses

-8r^2+(6r^2-19r-7)+28r

We get rid of parentheses

-8r^2+6r^2-19r+28r-7

We add all the numbers together, and all the variables

-2r^2+9r-7

Back to the equation:

+(-2r^2+9r-7)


We multiply parentheses

(-2r^2+9r-7)+6r+=0

We get rid of parentheses

-2r^2+9r+6r-7+=0

We add all the numbers together, and all the variables

-2r^2+15r=0

a = -2; b = 15; c = 0;
Δ = b2-4ac
Δ = 152-4·(-2)·0
Δ = 225
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$r_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$r_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$\sqrt{\Delta}=\sqrt{225}=15$
$r_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(15)-15}{2*-2}=\frac{-30}{-4}
=7+1/2
$
$r_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(15)+15}{2*-2}=\frac{0}{-4}
=0
$