3/4x+4x-6=-1/4x+42

Solution for 3/4x+4x-6=-1/4x+42 equation:

3/4x+4x-6=-1/4x+42

We move all terms to the left:

3/4x+4x-6-(-1/4x+42)=0


Domain of the equation: 4x!=0

x!=0/4

x!=0

x∈R



Domain of the equation: 4x+42)!=0

x∈R


We add all the numbers together, and all the variables

4x+3/4x-(-1/4x+42)-6=0

We get rid of parentheses

4x+3/4x+1/4x-42-6=0

We multiply all the terms by the denominator


4x*4x-42*4x-6*4x+3+1=0

We add all the numbers together, and all the variables

4x*4x-42*4x-6*4x+4=0

Wy multiply elements

16x^2-168x-24x+4=0

We add all the numbers together, and all the variables

16x^2-192x+4=0

a = 16; b = -192; c = +4;
Δ = b2-4ac
Δ = -1922-4·16·4
Δ = 36608
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:

$\sqrt{\Delta}=\sqrt{36608}=\sqrt{256*143}=\sqrt{256}*\sqrt{143}=16\sqrt{143}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-192)-16\sqrt{143}}{2*16}=\frac{192-16\sqrt{143}}{32}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-192)+16\sqrt{143}}{2*16}=\frac{192+16\sqrt{143}}{32}
$