1/2*x+5/3*x=52

Solution for 1/2*x+5/3*x=52 equation:

1/2x+5/3x=52

We move all terms to the left:

1/2x+5/3x-(52)=0


Domain of the equation: 2x!=0

x!=0/2

x!=0

x∈R



Domain of the equation: 3x!=0

x!=0/3

x!=0

x∈R


We calculate fractions


3x/6x^2+10x/6x^2-52=0

We multiply all the terms by the denominator


3x+10x-52*6x^2=0

We add all the numbers together, and all the variables

13x-52*6x^2=0

Wy multiply elements

-312x^2+13x=0

a = -312; b = 13; c = 0;
Δ = b2-4ac
Δ = 132-4·(-312)·0
Δ = 169
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$\sqrt{\Delta}=\sqrt{169}=13$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(13)-13}{2*-312}=\frac{-26}{-624}
=1/24
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(13)+13}{2*-312}=\frac{0}{-624}
=0
$