(5x+9)/(2x-5)=x

Solution for (5x+9)/(2x-5)=x equation:

(5x+9)/(2x-5)=x

We move all terms to the left:

(5x+9)/(2x-5)-(x)=0


Domain of the equation: (2x-5)!=0

We move all terms containing x to the left, all other terms to the right

2x!=5

x!=5/2

x!=2+1/2

x∈R


We add all the numbers together, and all the variables

-1x+(5x+9)/(2x-5)=0

We multiply all the terms by the denominator


-1x*(2x-5)+(5x+9)=0

We multiply parentheses

-2x^2+5x+(5x+9)=0

We get rid of parentheses

-2x^2+5x+5x+9=0

We add all the numbers together, and all the variables

-2x^2+10x+9=0

a = -2; b = 10; c = +9;
Δ = b2-4ac
Δ = 102-4·(-2)·9
Δ = 172
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:

$\sqrt{\Delta}=\sqrt{172}=\sqrt{4*43}=\sqrt{4}*\sqrt{43}=2\sqrt{43}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(10)-2\sqrt{43}}{2*-2}=\frac{-10-2\sqrt{43}}{-4}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(10)+2\sqrt{43}}{2*-2}=\frac{-10+2\sqrt{43}}{-4}
$