2-1/4x=3x+16

Solution for 2-1/4x=3x+16 equation:

2-1/4x=3x+16

We move all terms to the left:

2-1/4x-(3x+16)=0


Domain of the equation: 4x!=0

x!=0/4

x!=0

x∈R


We get rid of parentheses

-1/4x-3x-16+2=0

We multiply all the terms by the denominator


-3x*4x-16*4x+2*4x-1=0

Wy multiply elements

-12x^2-64x+8x-1=0

We add all the numbers together, and all the variables

-12x^2-56x-1=0

a = -12; b = -56; c = -1;
Δ = b2-4ac
Δ = -562-4·(-12)·(-1)
Δ = 3088
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:

$\sqrt{\Delta}=\sqrt{3088}=\sqrt{16*193}=\sqrt{16}*\sqrt{193}=4\sqrt{193}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-56)-4\sqrt{193}}{2*-12}=\frac{56-4\sqrt{193}}{-24}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-56)+4\sqrt{193}}{2*-12}=\frac{56+4\sqrt{193}}{-24}
$