10+10q=100/q+10+5q

Solution for 10+10q=100/q+10+5q equation:

10+10q=100/q+10+5q

We move all terms to the left:

10+10q-(100/q+10+5q)=0


Domain of the equation: q+10+5q)!=0

We move all terms containing q to the left, all other terms to the right

q+5q)!=-10

q∈R


We add all the numbers together, and all the variables

10q-(5q+100/q+10)+10=0

We get rid of parentheses

10q-5q-100/q-10+10=0

We multiply all the terms by the denominator


10q*q-5q*q-10*q+10*q-100=0

We add all the numbers together, and all the variables

10q*q-5q*q-100=0

Wy multiply elements

10q^2-5q^2-100=0

We add all the numbers together, and all the variables

5q^2-100=0

a = 5; b = 0; c = -100;
Δ = b2-4ac
Δ = 02-4·5·(-100)
Δ = 2000
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$q_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$q_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{2000}=\sqrt{400*5}=\sqrt{400}*\sqrt{5}=20\sqrt{5}$
$q_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(0)-20\sqrt{5}}{2*5}=\frac{0-20\sqrt{5}}{10}
=-\frac{20\sqrt{5}}{10}
=-2\sqrt{5}
$
$q_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(0)+20\sqrt{5}}{2*5}=\frac{0+20\sqrt{5}}{10}
=\frac{20\sqrt{5}}{10}
=2\sqrt{5}
$