1/2x-5=x+4

Solution for 1/2x-5=x+4 equation:

1/2x-5=x+4

We move all terms to the left:

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


Domain of the equation: 2x!=0

x!=0/2

x!=0

x∈R


We get rid of parentheses

1/2x-x-4-5=0

We multiply all the terms by the denominator


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

Wy multiply elements

-2x^2-8x-10x+1=0

We add all the numbers together, and all the variables

-2x^2-18x+1=0

a = -2; b = -18; c = +1;
Δ = b2-4ac
Δ = -182-4·(-2)·1
Δ = 332
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:

$\sqrt{\Delta}=\sqrt{332}=\sqrt{4*83}=\sqrt{4}*\sqrt{83}=2\sqrt{83}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-18)-2\sqrt{83}}{2*-2}=\frac{18-2\sqrt{83}}{-4}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-18)+2\sqrt{83}}{2*-2}=\frac{18+2\sqrt{83}}{-4}
$