x+(x-5)+1/2x=75

Solution for x+(x-5)+1/2x=75 equation:

x+(x-5)+1/2x=75

We move all terms to the left:

x+(x-5)+1/2x-(75)=0


Domain of the equation: 2x!=0

x!=0/2

x!=0

x∈R


We get rid of parentheses

x+x+1/2x-5-75=0

We multiply all the terms by the denominator


x*2x+x*2x-5*2x-75*2x+1=0

Wy multiply elements

2x^2+2x^2-10x-150x+1=0

We add all the numbers together, and all the variables

4x^2-160x+1=0

a = 4; b = -160; c = +1;
Δ = b2-4ac
Δ = -1602-4·4·1
Δ = 25584
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:

$\sqrt{\Delta}=\sqrt{25584}=\sqrt{16*1599}=\sqrt{16}*\sqrt{1599}=4\sqrt{1599}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-160)-4\sqrt{1599}}{2*4}=\frac{160-4\sqrt{1599}}{8}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-160)+4\sqrt{1599}}{2*4}=\frac{160+4\sqrt{1599}}{8}
$