1/2x+3x+4x=1000

Solution for 1/2x+3x+4x=1000 equation:

1/2x+3x+4x=1000

We move all terms to the left:

1/2x+3x+4x-(1000)=0


Domain of the equation: 2x!=0

x!=0/2

x!=0

x∈R


We add all the numbers together, and all the variables

7x+1/2x-1000=0

We multiply all the terms by the denominator


7x*2x-1000*2x+1=0

Wy multiply elements

14x^2-2000x+1=0

a = 14; b = -2000; c = +1;
Δ = b2-4ac
Δ = -20002-4·14·1
Δ = 3999944
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:

$\sqrt{\Delta}=\sqrt{3999944}=\sqrt{4*999986}=\sqrt{4}*\sqrt{999986}=2\sqrt{999986}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-2000)-2\sqrt{999986}}{2*14}=\frac{2000-2\sqrt{999986}}{28}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-2000)+2\sqrt{999986}}{2*14}=\frac{2000+2\sqrt{999986}}{28}
$