(1)/(4)x-53x+2=-5x+10

Solution for (1)/(4)x-53x+2=-5x+10 equation:

(1)/(4)x-53x+2=-5x+10

We move all terms to the left:

(1)/(4)x-53x+2-(-5x+10)=0


Domain of the equation: 4x!=0

x!=0/4

x!=0

x∈R


We add all the numbers together, and all the variables

-53x+1/4x-(-5x+10)+2=0

We get rid of parentheses

-53x+1/4x+5x-10+2=0

We multiply all the terms by the denominator


-53x*4x+5x*4x-10*4x+2*4x+1=0

Wy multiply elements

-212x^2+20x^2-40x+8x+1=0

We add all the numbers together, and all the variables

-192x^2-32x+1=0

a = -192; b = -32; c = +1;
Δ = b2-4ac
Δ = -322-4·(-192)·1
Δ = 1792
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:

$\sqrt{\Delta}=\sqrt{1792}=\sqrt{256*7}=\sqrt{256}*\sqrt{7}=16\sqrt{7}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-32)-16\sqrt{7}}{2*-192}=\frac{32-16\sqrt{7}}{-384}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-32)+16\sqrt{7}}{2*-192}=\frac{32+16\sqrt{7}}{-384}
$