1/4x-2x+3=x+2-7

Solution for 1/4x-2x+3=x+2-7 equation:

1/4x-2x+3=x+2-7

We move all terms to the left:

1/4x-2x+3-(x+2-7)=0


Domain of the equation: 4x!=0

x!=0/4

x!=0

x∈R


We add all the numbers together, and all the variables

1/4x-2x-(x-5)+3=0

We add all the numbers together, and all the variables

-2x+1/4x-(x-5)+3=0

We get rid of parentheses

-2x+1/4x-x+5+3=0

We multiply all the terms by the denominator


-2x*4x-x*4x+5*4x+3*4x+1=0

Wy multiply elements

-8x^2-4x^2+20x+12x+1=0

We add all the numbers together, and all the variables

-12x^2+32x+1=0

a = -12; b = 32; c = +1;
Δ = b2-4ac
Δ = 322-4·(-12)·1
Δ = 1072
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:

$\sqrt{\Delta}=\sqrt{1072}=\sqrt{16*67}=\sqrt{16}*\sqrt{67}=4\sqrt{67}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(32)-4\sqrt{67}}{2*-12}=\frac{-32-4\sqrt{67}}{-24}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(32)+4\sqrt{67}}{2*-12}=\frac{-32+4\sqrt{67}}{-24}
$