4(0.5n-3)n=-0.25(12-8n)

Solution for 4(0.5n-3)n=-0.25(12-8n) equation:

4(0.5n-3)n=-0.25(12-8n)

We move all terms to the left:

4(0.5n-3)n-(-0.25(12-8n))=0

We add all the numbers together, and all the variables

4(0.5n-3)n-(-0.25(-8n+12))=0

We multiply parentheses

0n^2-12n-(-0.25(-8n+12))=0


We calculate terms in parentheses: -(-0.25(-8n+12)), so:

-0.25(-8n+12)

We multiply parentheses

2n-3

Back to the equation:

-(2n-3)


We add all the numbers together, and all the variables

n^2-12n-(2n-3)=0

We get rid of parentheses

n^2-12n-2n+3=0

We add all the numbers together, and all the variables

n^2-14n+3=0

a = 1; b = -14; c = +3;
Δ = b2-4ac
Δ = -142-4·1·3
Δ = 184
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{184}=\sqrt{4*46}=\sqrt{4}*\sqrt{46}=2\sqrt{46}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-14)-2\sqrt{46}}{2*1}=\frac{14-2\sqrt{46}}{2}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-14)+2\sqrt{46}}{2*1}=\frac{14+2\sqrt{46}}{2}
$