6.5f=14-1/2f

Solution for 6.5f=14-1/2f equation:

6.5f=14-1/2f

We move all terms to the left:

6.5f-(14-1/2f)=0


Domain of the equation: 2f)!=0

f!=0/1

f!=0

f∈R


We add all the numbers together, and all the variables

6.5f-(-1/2f+14)=0

We get rid of parentheses

6.5f+1/2f-14=0

We multiply all the terms by the denominator


(6.5f)*2f-14*2f+1=0

We add all the numbers together, and all the variables

(+6.5f)*2f-14*2f+1=0

We multiply parentheses

12f^2-14*2f+1=0

Wy multiply elements

12f^2-28f+1=0

a = 12; b = -28; c = +1;
Δ = b2-4ac
Δ = -282-4·12·1
Δ = 736
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$f_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$f_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{736}=\sqrt{16*46}=\sqrt{16}*\sqrt{46}=4\sqrt{46}$
$f_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-28)-4\sqrt{46}}{2*12}=\frac{28-4\sqrt{46}}{24}
$
$f_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-28)+4\sqrt{46}}{2*12}=\frac{28+4\sqrt{46}}{24}
$