4x-3=1/4x+19

Solution for 4x-3=1/4x+19 equation:

4x-3=1/4x+19

We move all terms to the left:

4x-3-(1/4x+19)=0


Domain of the equation: 4x+19)!=0

x∈R


We get rid of parentheses

4x-1/4x-19-3=0

We multiply all the terms by the denominator


4x*4x-19*4x-3*4x-1=0

Wy multiply elements

16x^2-76x-12x-1=0

We add all the numbers together, and all the variables

16x^2-88x-1=0

a = 16; b = -88; c = -1;
Δ = b2-4ac
Δ = -882-4·16·(-1)
Δ = 7808
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:

$\sqrt{\Delta}=\sqrt{7808}=\sqrt{64*122}=\sqrt{64}*\sqrt{122}=8\sqrt{122}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-88)-8\sqrt{122}}{2*16}=\frac{88-8\sqrt{122}}{32}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-88)+8\sqrt{122}}{2*16}=\frac{88+8\sqrt{122}}{32}
$