1/4x-3.5=3x-8

Solution for 1/4x-3.5=3x-8 equation:

1/4x-3.5=3x-8

We move all terms to the left:

1/4x-3.5-(3x-8)=0


Domain of the equation: 4x!=0

x!=0/4

x!=0

x∈R


We get rid of parentheses

1/4x-3x+8-3.5=0

We multiply all the terms by the denominator


-3x*4x+8*4x-(3.5)*4x+1=0

We multiply parentheses

-3x*4x+8*4x-14x+1=0

Wy multiply elements

-12x^2+32x-14x+1=0

We add all the numbers together, and all the variables

-12x^2+18x+1=0

a = -12; b = 18; c = +1;
Δ = b2-4ac
Δ = 182-4·(-12)·1
Δ = 372
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:

$\sqrt{\Delta}=\sqrt{372}=\sqrt{4*93}=\sqrt{4}*\sqrt{93}=2\sqrt{93}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(18)-2\sqrt{93}}{2*-12}=\frac{-18-2\sqrt{93}}{-24}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(18)+2\sqrt{93}}{2*-12}=\frac{-18+2\sqrt{93}}{-24}
$