3/2x+3+4=4x-5/2x-7

Solution for 3/2x+3+4=4x-5/2x-7 equation:

3/2x+3+4=4x-5/2x-7

We move all terms to the left:

3/2x+3+4-(4x-5/2x-7)=0


Domain of the equation: 2x!=0

x!=0/2

x!=0

x∈R



Domain of the equation: 2x-7)!=0

x∈R


We add all the numbers together, and all the variables

3/2x-(4x-5/2x-7)+7=0

We get rid of parentheses

3/2x-4x+5/2x+7+7=0

We multiply all the terms by the denominator


-4x*2x+7*2x+7*2x+3+5=0

We add all the numbers together, and all the variables

-4x*2x+7*2x+7*2x+8=0

Wy multiply elements

-8x^2+14x+14x+8=0

We add all the numbers together, and all the variables

-8x^2+28x+8=0

a = -8; b = 28; c = +8;
Δ = b2-4ac
Δ = 282-4·(-8)·8
Δ = 1040
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:

$\sqrt{\Delta}=\sqrt{1040}=\sqrt{16*65}=\sqrt{16}*\sqrt{65}=4\sqrt{65}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(28)-4\sqrt{65}}{2*-8}=\frac{-28-4\sqrt{65}}{-16}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(28)+4\sqrt{65}}{2*-8}=\frac{-28+4\sqrt{65}}{-16}
$