65=31/4x+1+2x

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

65=31/4x+1+2x

We move all terms to the left:

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


Domain of the equation: 4x+1+2x)!=0

We move all terms containing x to the left, all other terms to the right

4x+2x)!=-1

x∈R


We add all the numbers together, and all the variables

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

We get rid of parentheses

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

We multiply all the terms by the denominator


-2x*4x-1*4x+65*4x-31=0

Wy multiply elements

-8x^2-4x+260x-31=0

We add all the numbers together, and all the variables

-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}
$