x(3-5/x)+7=x-2(x-2)

Solution for x(3-5/x)+7=x-2(x-2) equation:

x(3-5/x)+7=x-2(x-2)

We move all terms to the left:

x(3-5/x)+7-(x-2(x-2))=0


Domain of the equation: x)!=0

x!=0/1

x!=0

x∈R


We add all the numbers together, and all the variables

x(-5/x+3)-(x-2(x-2))+7=0

We multiply parentheses

-5x^2+3x-(x-2(x-2))+7=0


We calculate terms in parentheses: -(x-2(x-2)), so:

x-2(x-2)

We multiply parentheses

x-2x+4

We add all the numbers together, and all the variables

-1x+4

Back to the equation:

-(-1x+4)


We get rid of parentheses

-5x^2+3x+1x-4+7=0

We add all the numbers together, and all the variables

-5x^2+4x+3=0

a = -5; b = 4; c = +3;
Δ = b2-4ac
Δ = 42-4·(-5)·3
Δ = 76
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:

$\sqrt{\Delta}=\sqrt{76}=\sqrt{4*19}=\sqrt{4}*\sqrt{19}=2\sqrt{19}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(4)-2\sqrt{19}}{2*-5}=\frac{-4-2\sqrt{19}}{-10}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(4)+2\sqrt{19}}{2*-5}=\frac{-4+2\sqrt{19}}{-10}
$