x/x-6=x-7/3x-5

Solution for x/x-6=x-7/3x-5 equation:

x/x-6=x-7/3x-5

We move all terms to the left:

x/x-6-(x-7/3x-5)=0


Domain of the equation: x!=0

x∈R



Domain of the equation: 3x-5)!=0

x∈R


We get rid of parentheses

x/x-x+7/3x+5-6=0

Fractions to decimals

7/3x-x+5-6+1=0

We multiply all the terms by the denominator


-x*3x+5*3x-6*3x+1*3x+7=0

Wy multiply elements

-3x^2+15x-18x+3x+7=0

We add all the numbers together, and all the variables

-3x^2+7=0

a = -3; b = 0; c = +7;
Δ = b2-4ac
Δ = 02-4·(-3)·7
Δ = 84
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{84}=\sqrt{4*21}=\sqrt{4}*\sqrt{21}=2\sqrt{21}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(0)-2\sqrt{21}}{2*-3}=\frac{0-2\sqrt{21}}{-6}
=-\frac{2\sqrt{21}}{-6}
=-\frac{\sqrt{21}}{-3}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(0)+2\sqrt{21}}{2*-3}=\frac{0+2\sqrt{21}}{-6}
=\frac{2\sqrt{21}}{-6}
=\frac{\sqrt{21}}{-3}
$