(X+3)=3x+4/x-7

Solution for (X+3)=3x+4/x-7 equation:

(X+3)=3X+4/X-7

We move all terms to the left:

(X+3)-(3X+4/X-7)=0


Domain of the equation: X-7)!=0

X∈R


We get rid of parentheses

X-3X-4/X+3+7=0

We multiply all the terms by the denominator


X*X-3X*X+3*X+7*X-4=0

We add all the numbers together, and all the variables

10X+X*X-3X*X-4=0

Wy multiply elements

X^2-3X^2+10X-4=0

We add all the numbers together, and all the variables

-2X^2+10X-4=0

a = -2; b = 10; c = -4;
Δ = b2-4ac
Δ = 102-4·(-2)·(-4)
Δ = 68
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:

$\sqrt{\Delta}=\sqrt{68}=\sqrt{4*17}=\sqrt{4}*\sqrt{17}=2\sqrt{17}$
$X_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(10)-2\sqrt{17}}{2*-2}=\frac{-10-2\sqrt{17}}{-4}
$
$X_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(10)+2\sqrt{17}}{2*-2}=\frac{-10+2\sqrt{17}}{-4}
$