31/4x+1+2x=65

Solution for 31/4x+1+2x=65 equation:

31/4x+1+2x=65

We move all terms to the left:

31/4x+1+2x-(65)=0


Domain of the equation: 4x!=0

x!=0/4

x!=0

x∈R


We add all the numbers together, and all the variables

2x+31/4x-64=0

We multiply all the terms by the denominator


2x*4x-64*4x+31=0

Wy multiply elements

8x^2-256x+31=0

a = 8; b = -256; c = +31;
Δ = b2-4ac
Δ = -2562-4·8·31
Δ = 64544
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:

$\sqrt{\Delta}=\sqrt{64544}=\sqrt{16*4034}=\sqrt{16}*\sqrt{4034}=4\sqrt{4034}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-256)-4\sqrt{4034}}{2*8}=\frac{256-4\sqrt{4034}}{16}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-256)+4\sqrt{4034}}{2*8}=\frac{256+4\sqrt{4034}}{16}
$