1/4x-3=x+6

Solution for 1/4x-3=x+6 equation:

1/4x-3=x+6

We move all terms to the left:

1/4x-3-(x+6)=0


Domain of the equation: 4x!=0

x!=0/4

x!=0

x∈R


We get rid of parentheses

1/4x-x-6-3=0

We multiply all the terms by the denominator


-x*4x-6*4x-3*4x+1=0

Wy multiply elements

-4x^2-24x-12x+1=0

We add all the numbers together, and all the variables

-4x^2-36x+1=0

a = -4; b = -36; c = +1;
Δ = b2-4ac
Δ = -362-4·(-4)·1
Δ = 1312
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:

$\sqrt{\Delta}=\sqrt{1312}=\sqrt{16*82}=\sqrt{16}*\sqrt{82}=4\sqrt{82}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-36)-4\sqrt{82}}{2*-4}=\frac{36-4\sqrt{82}}{-8}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-36)+4\sqrt{82}}{2*-4}=\frac{36+4\sqrt{82}}{-8}
$