5x+1/2x=45-x

Solution for 5x+1/2x=45-x equation:

5x+1/2x=45-x

We move all terms to the left:

5x+1/2x-(45-x)=0


Domain of the equation: 2x!=0

x!=0/2

x!=0

x∈R


We add all the numbers together, and all the variables

5x+1/2x-(-1x+45)=0

We get rid of parentheses

5x+1/2x+1x-45=0

We multiply all the terms by the denominator


5x*2x+1x*2x-45*2x+1=0

Wy multiply elements

10x^2+2x^2-90x+1=0

We add all the numbers together, and all the variables

12x^2-90x+1=0

a = 12; b = -90; c = +1;
Δ = b2-4ac
Δ = -902-4·12·1
Δ = 8052
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:

$\sqrt{\Delta}=\sqrt{8052}=\sqrt{4*2013}=\sqrt{4}*\sqrt{2013}=2\sqrt{2013}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-90)-2\sqrt{2013}}{2*12}=\frac{90-2\sqrt{2013}}{24}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-90)+2\sqrt{2013}}{2*12}=\frac{90+2\sqrt{2013}}{24}
$