2x=31/4x+1

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

2x=31/4x+1

We move all terms to the left:

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


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

x∈R


We get rid of parentheses

2x-31/4x-1=0

We multiply all the terms by the denominator


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

Wy multiply elements

8x^2-4x-31=0

a = 8; b = -4; c = -31;
Δ = b2-4ac
Δ = -42-4·8·(-31)
Δ = 1008
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:

$\sqrt{\Delta}=\sqrt{1008}=\sqrt{144*7}=\sqrt{144}*\sqrt{7}=12\sqrt{7}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-4)-12\sqrt{7}}{2*8}=\frac{4-12\sqrt{7}}{16}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-4)+12\sqrt{7}}{2*8}=\frac{4+12\sqrt{7}}{16}
$