5u+3/4u=54-u

Solution for 5u+3/4u=54-u equation:

5u+3/4u=54-u

We move all terms to the left:

5u+3/4u-(54-u)=0


Domain of the equation: 4u!=0

u!=0/4

u!=0

u∈R


We add all the numbers together, and all the variables

5u+3/4u-(-1u+54)=0

We get rid of parentheses

5u+3/4u+1u-54=0

We multiply all the terms by the denominator


5u*4u+1u*4u-54*4u+3=0

Wy multiply elements

20u^2+4u^2-216u+3=0

We add all the numbers together, and all the variables

24u^2-216u+3=0

a = 24; b = -216; c = +3;
Δ = b2-4ac
Δ = -2162-4·24·3
Δ = 46368
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$u_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$u_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{46368}=\sqrt{144*322}=\sqrt{144}*\sqrt{322}=12\sqrt{322}$
$u_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-216)-12\sqrt{322}}{2*24}=\frac{216-12\sqrt{322}}{48}
$
$u_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-216)+12\sqrt{322}}{2*24}=\frac{216+12\sqrt{322}}{48}
$