9+1/2x=5x-10

Solution for 9+1/2x=5x-10 equation:

9+1/2x=5x-10

We move all terms to the left:

9+1/2x-(5x-10)=0


Domain of the equation: 2x!=0

x!=0/2

x!=0

x∈R


We get rid of parentheses

1/2x-5x+10+9=0

We multiply all the terms by the denominator


-5x*2x+10*2x+9*2x+1=0

Wy multiply elements

-10x^2+20x+18x+1=0

We add all the numbers together, and all the variables

-10x^2+38x+1=0

a = -10; b = 38; c = +1;
Δ = b2-4ac
Δ = 382-4·(-10)·1
Δ = 1484
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:

$\sqrt{\Delta}=\sqrt{1484}=\sqrt{4*371}=\sqrt{4}*\sqrt{371}=2\sqrt{371}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(38)-2\sqrt{371}}{2*-10}=\frac{-38-2\sqrt{371}}{-20}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(38)+2\sqrt{371}}{2*-10}=\frac{-38+2\sqrt{371}}{-20}
$