3/4x+2x-3=-1/4x+22

Solution for 3/4x+2x-3=-1/4x+22 equation:

3/4x+2x-3=-1/4x+22

We move all terms to the left:

3/4x+2x-3-(-1/4x+22)=0


Domain of the equation: 4x!=0

x!=0/4

x!=0

x∈R



Domain of the equation: 4x+22)!=0

x∈R


We add all the numbers together, and all the variables

2x+3/4x-(-1/4x+22)-3=0

We get rid of parentheses

2x+3/4x+1/4x-22-3=0

We multiply all the terms by the denominator


2x*4x-22*4x-3*4x+3+1=0

We add all the numbers together, and all the variables

2x*4x-22*4x-3*4x+4=0

Wy multiply elements

8x^2-88x-12x+4=0

We add all the numbers together, and all the variables

8x^2-100x+4=0

a = 8; b = -100; c = +4;
Δ = b2-4ac
Δ = -1002-4·8·4
Δ = 9872
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:

$\sqrt{\Delta}=\sqrt{9872}=\sqrt{16*617}=\sqrt{16}*\sqrt{617}=4\sqrt{617}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-100)-4\sqrt{617}}{2*8}=\frac{100-4\sqrt{617}}{16}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-100)+4\sqrt{617}}{2*8}=\frac{100+4\sqrt{617}}{16}
$