5(x+6)-15(2-x)/3x-1=10

Solution for 5(x+6)-15(2-x)/3x-1=10 equation:

5(x+6)-15(2-x)/3x-1=10

We move all terms to the left:

5(x+6)-15(2-x)/3x-1-(10)=0


Domain of the equation: 3x!=0

x!=0/3

x!=0

x∈R


We add all the numbers together, and all the variables

5(x+6)-15(-1x+2)/3x-1-10=0

We add all the numbers together, and all the variables

5(x+6)-15(-1x+2)/3x-11=0

We multiply parentheses

5x-15(-1x+2)/3x+30-11=0

We multiply all the terms by the denominator


5x*3x-15(-1x+2)+30*3x-11*3x=0

We multiply parentheses

5x*3x+15x+30*3x-11*3x-30=0

Wy multiply elements

15x^2+15x+90x-33x-30=0

We add all the numbers together, and all the variables

15x^2+72x-30=0

a = 15; b = 72; c = -30;
Δ = b2-4ac
Δ = 722-4·15·(-30)
Δ = 6984
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{6984}=\sqrt{36*194}=\sqrt{36}*\sqrt{194}=6\sqrt{194}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(72)-6\sqrt{194}}{2*15}=\frac{-72-6\sqrt{194}}{30}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(72)+6\sqrt{194}}{2*15}=\frac{-72+6\sqrt{194}}{30}
$