3/4w+67/16w+9=24-1/16w

Solution for 3/4w+67/16w+9=24-1/16w equation:

3/4w+67/16w+9=24-1/16w

We move all terms to the left:

3/4w+67/16w+9-(24-1/16w)=0


Domain of the equation: 4w!=0

w!=0/4

w!=0

w∈R



Domain of the equation: 16w!=0

w!=0/16

w!=0

w∈R



Domain of the equation: 16w)!=0

w!=0/1

w!=0

w∈R


We add all the numbers together, and all the variables

3/4w+67/16w-(-1/16w+24)+9=0

We get rid of parentheses

3/4w+67/16w+1/16w-24+9=0

We calculate fractions


48w/64w^2+(4w+67)/64w^2-24+9=0

We add all the numbers together, and all the variables

48w/64w^2+(4w+67)/64w^2-15=0

We multiply all the terms by the denominator


48w+(4w+67)-15*64w^2=0

Wy multiply elements

-960w^2+48w+(4w+67)=0

We get rid of parentheses

-960w^2+48w+4w+67=0

We add all the numbers together, and all the variables

-960w^2+52w+67=0

a = -960; b = 52; c = +67;
Δ = b2-4ac
Δ = 522-4·(-960)·67
Δ = 259984
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$w_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$w_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{259984}=\sqrt{16*16249}=\sqrt{16}*\sqrt{16249}=4\sqrt{16249}$
$w_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(52)-4\sqrt{16249}}{2*-960}=\frac{-52-4\sqrt{16249}}{-1920}
$
$w_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(52)+4\sqrt{16249}}{2*-960}=\frac{-52+4\sqrt{16249}}{-1920}
$