31/2x+2=2x-7

Solution for 31/2x+2=2x-7 equation:

31/2x+2=2x-7

We move all terms to the left:

31/2x+2-(2x-7)=0


Domain of the equation: 2x!=0

x!=0/2

x!=0

x∈R


We get rid of parentheses

31/2x-2x+7+2=0

We multiply all the terms by the denominator


-2x*2x+7*2x+2*2x+31=0

Wy multiply elements

-4x^2+14x+4x+31=0

We add all the numbers together, and all the variables

-4x^2+18x+31=0

a = -4; b = 18; c = +31;
Δ = b2-4ac
Δ = 182-4·(-4)·31
Δ = 820
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:

$\sqrt{\Delta}=\sqrt{820}=\sqrt{4*205}=\sqrt{4}*\sqrt{205}=2\sqrt{205}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(18)-2\sqrt{205}}{2*-4}=\frac{-18-2\sqrt{205}}{-8}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(18)+2\sqrt{205}}{2*-4}=\frac{-18+2\sqrt{205}}{-8}
$