5/4x-6=2/x-9

Solution for 5/4x-6=2/x-9 equation:

5/4x-6=2/x-9

We move all terms to the left:

5/4x-6-(2/x-9)=0


Domain of the equation: 4x!=0

x!=0/4

x!=0

x∈R



Domain of the equation: x-9)!=0

x∈R


We get rid of parentheses

5/4x-2/x+9-6=0

We calculate fractions


5x/4x^2+(-8x)/4x^2+9-6=0

We add all the numbers together, and all the variables

5x/4x^2+(-8x)/4x^2+3=0

We multiply all the terms by the denominator


5x+(-8x)+3*4x^2=0

Wy multiply elements

12x^2+5x+(-8x)=0

We get rid of parentheses

12x^2+5x-8x=0

We add all the numbers together, and all the variables

12x^2-3x=0

a = 12; b = -3; c = 0;
Δ = b2-4ac
Δ = -32-4·12·0
Δ = 9
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$\sqrt{\Delta}=\sqrt{9}=3$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-3)-3}{2*12}=\frac{0}{24}
=0
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-3)+3}{2*12}=\frac{6}{24}
=1/4
$