1/2x-6.5=12-x

Solution for 1/2x-6.5=12-x equation:

1/2x-6.5=12-x

We move all terms to the left:

1/2x-6.5-(12-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

1/2x-(-1x+12)-6.5=0

We get rid of parentheses

1/2x+1x-12-6.5=0

We multiply all the terms by the denominator


1x*2x-12*2x-(6.5)*2x+1=0

We multiply parentheses

1x*2x-12*2x-13x+1=0

Wy multiply elements

2x^2-24x-13x+1=0

We add all the numbers together, and all the variables

2x^2-37x+1=0

a = 2; b = -37; c = +1;
Δ = b2-4ac
Δ = -372-4·2·1
Δ = 1361
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-37)-\sqrt{1361}}{2*2}=\frac{37-\sqrt{1361}}{4}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-37)+\sqrt{1361}}{2*2}=\frac{37+\sqrt{1361}}{4}
$