X/5-3=x/2,x2-x

Solution for X/5-3=x/2,x2-x equation:

X/5-3=X/2.X^2-X

We move all terms to the left:

X/5-3-(X/2.X^2-X)=0


Domain of the equation: 2.X^2-X)!=0

X∈R


We get rid of parentheses

X/5-X/2.X^2+X-3=0

We calculate fractions


2X^2/10X+X+(-5X)/10X-3=0

We multiply all the terms by the denominator


2X^2+X*10X+(-5X)-3*10X=0

Wy multiply elements

2X^2+10X^2+(-5X)-30X=0

We get rid of parentheses

2X^2+10X^2-5X-30X=0

We add all the numbers together, and all the variables

12X^2-35X=0

a = 12; b = -35; c = 0;
Δ = b2-4ac
Δ = -352-4·12·0
Δ = 1225
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$X_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$X_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$\sqrt{\Delta}=\sqrt{1225}=35$
$X_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-35)-35}{2*12}=\frac{0}{24}
=0
$
$X_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-35)+35}{2*12}=\frac{70}{24}
=2+11/12
$