X(3-1/x)+1=3(x+1)-7

Solution for X(3-1/x)+1=3(x+1)-7 equation:

X(3-1/X)+1=3(X+1)-7

We move all terms to the left:

X(3-1/X)+1-(3(X+1)-7)=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(-1/X+3)-(3(X+1)-7)+1=0

We multiply parentheses

-1X^2+3X-(3(X+1)-7)+1=0


We calculate terms in parentheses: -(3(X+1)-7), so:

3(X+1)-7

We multiply parentheses

3X+3-7

We add all the numbers together, and all the variables

3X-4

Back to the equation:

-(3X-4)


We get rid of parentheses

-1X^2+3X-3X+4+1=0

We add all the numbers together, and all the variables

-1X^2+5=0

a = -1; b = 0; c = +5;
Δ = b2-4ac
Δ = 02-4·(-1)·5
Δ = 20
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:

$\sqrt{\Delta}=\sqrt{20}=\sqrt{4*5}=\sqrt{4}*\sqrt{5}=2\sqrt{5}$
$X_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(0)-2\sqrt{5}}{2*-1}=\frac{0-2\sqrt{5}}{-2}
=-\frac{2\sqrt{5}}{-2}
=-\frac{\sqrt{5}}{-1}
$
$X_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(0)+2\sqrt{5}}{2*-1}=\frac{0+2\sqrt{5}}{-2}
=\frac{2\sqrt{5}}{-2}
=\frac{\sqrt{5}}{-1}
$