4-1/2x=5x-3

Solution for 4-1/2x=5x-3 equation:

4-1/2x=5x-3

We move all terms to the left:

4-1/2x-(5x-3)=0


Domain of the equation: 2x!=0

x!=0/2

x!=0

x∈R


We get rid of parentheses

-1/2x-5x+3+4=0

We multiply all the terms by the denominator


-5x*2x+3*2x+4*2x-1=0

Wy multiply elements

-10x^2+6x+8x-1=0

We add all the numbers together, and all the variables

-10x^2+14x-1=0

a = -10; b = 14; c = -1;
Δ = b2-4ac
Δ = 142-4·(-10)·(-1)
Δ = 156
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:

$\sqrt{\Delta}=\sqrt{156}=\sqrt{4*39}=\sqrt{4}*\sqrt{39}=2\sqrt{39}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(14)-2\sqrt{39}}{2*-10}=\frac{-14-2\sqrt{39}}{-20}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(14)+2\sqrt{39}}{2*-10}=\frac{-14+2\sqrt{39}}{-20}
$