(1/2x)+(31/4x+4)+(21/2x-10)=180

Solution for (1/2x)+(31/4x+4)+(21/2x-10)=180 equation:

(1/2x)+(31/4x+4)+(21/2x-10)=180

We move all terms to the left:

(1/2x)+(31/4x+4)+(21/2x-10)-(180)=0


Domain of the equation: 2x)!=0

x!=0/1

x!=0

x∈R



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

x∈R



Domain of the equation: 2x-10)!=0

x∈R


We add all the numbers together, and all the variables

(+1/2x)+(31/4x+4)+(21/2x-10)-180=0

We get rid of parentheses

1/2x+31/4x+21/2x+4-10-180=0

We calculate fractions


(84x+1)/8x^2+62x/8x^2+4-10-180=0

We add all the numbers together, and all the variables

(84x+1)/8x^2+62x/8x^2-186=0

We multiply all the terms by the denominator


(84x+1)+62x-186*8x^2=0

We add all the numbers together, and all the variables

62x+(84x+1)-186*8x^2=0

Wy multiply elements

-1488x^2+62x+(84x+1)=0

We get rid of parentheses

-1488x^2+62x+84x+1=0

We add all the numbers together, and all the variables

-1488x^2+146x+1=0

a = -1488; b = 146; c = +1;
Δ = b2-4ac
Δ = 1462-4·(-1488)·1
Δ = 27268
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:

$\sqrt{\Delta}=\sqrt{27268}=\sqrt{4*6817}=\sqrt{4}*\sqrt{6817}=2\sqrt{6817}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(146)-2\sqrt{6817}}{2*-1488}=\frac{-146-2\sqrt{6817}}{-2976}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(146)+2\sqrt{6817}}{2*-1488}=\frac{-146+2\sqrt{6817}}{-2976}
$