1/4x+6=x-5

Solution for 1/4x+6=x-5 equation:

1/4x+6=x-5

We move all terms to the left:

1/4x+6-(x-5)=0


Domain of the equation: 4x!=0

x!=0/4

x!=0

x∈R


We get rid of parentheses

1/4x-x+5+6=0

We multiply all the terms by the denominator


-x*4x+5*4x+6*4x+1=0

Wy multiply elements

-4x^2+20x+24x+1=0

We add all the numbers together, and all the variables

-4x^2+44x+1=0

a = -4; b = 44; c = +1;
Δ = b2-4ac
Δ = 442-4·(-4)·1
Δ = 1952
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:

$\sqrt{\Delta}=\sqrt{1952}=\sqrt{16*122}=\sqrt{16}*\sqrt{122}=4\sqrt{122}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(44)-4\sqrt{122}}{2*-4}=\frac{-44-4\sqrt{122}}{-8}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(44)+4\sqrt{122}}{2*-4}=\frac{-44+4\sqrt{122}}{-8}
$