2/3x-5=-4x+9

Solution for 2/3x-5=-4x+9 equation:

2/3x-5=-4x+9

We move all terms to the left:

2/3x-5-(-4x+9)=0


Domain of the equation: 3x!=0

x!=0/3

x!=0

x∈R


We get rid of parentheses

2/3x+4x-9-5=0

We multiply all the terms by the denominator


4x*3x-9*3x-5*3x+2=0

Wy multiply elements

12x^2-27x-15x+2=0

We add all the numbers together, and all the variables

12x^2-42x+2=0

a = 12; b = -42; c = +2;
Δ = b2-4ac
Δ = -422-4·12·2
Δ = 1668
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:

$\sqrt{\Delta}=\sqrt{1668}=\sqrt{4*417}=\sqrt{4}*\sqrt{417}=2\sqrt{417}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-42)-2\sqrt{417}}{2*12}=\frac{42-2\sqrt{417}}{24}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-42)+2\sqrt{417}}{2*12}=\frac{42+2\sqrt{417}}{24}
$