2/3x-5=10x+7

Solution for 2/3x-5=10x+7 equation:

2/3x-5=10x+7

We move all terms to the left:

2/3x-5-(10x+7)=0


Domain of the equation: 3x!=0

x!=0/3

x!=0

x∈R


We get rid of parentheses

2/3x-10x-7-5=0

We multiply all the terms by the denominator


-10x*3x-7*3x-5*3x+2=0

Wy multiply elements

-30x^2-21x-15x+2=0

We add all the numbers together, and all the variables

-30x^2-36x+2=0

a = -30; b = -36; c = +2;
Δ = b2-4ac
Δ = -362-4·(-30)·2
Δ = 1536
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:

$\sqrt{\Delta}=\sqrt{1536}=\sqrt{256*6}=\sqrt{256}*\sqrt{6}=16\sqrt{6}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-36)-16\sqrt{6}}{2*-30}=\frac{36-16\sqrt{6}}{-60}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-36)+16\sqrt{6}}{2*-30}=\frac{36+16\sqrt{6}}{-60}
$