(3X-1)2+(2X+3)2=5(x-2)+(x-1)(5x+2)

Solution for (3X-1)2+(2X+3)2=5(x-2)+(x-1)(5x+2) equation:

(3X-1)2+(2X+3)2=5(X-2)+(X-1)(5X+2)

We move all terms to the left:

(3X-1)2+(2X+3)2-(5(X-2)+(X-1)(5X+2))=0

We multiply parentheses

6X+4X-(5(X-2)+(X-1)(5X+2))-2+6=0

We multiply parentheses ..

-(5(X-2)+(+5X^2+2X-5X-2))+6X+4X-2+6=0


We calculate terms in parentheses: -(5(X-2)+(+5X^2+2X-5X-2)), so:

5(X-2)+(+5X^2+2X-5X-2)

determiningTheFunctionDomain
(+5X^2+2X-5X-2)+5(X-2)

We multiply parentheses

(+5X^2+2X-5X-2)+5X-10

We get rid of parentheses

5X^2+2X-5X+5X-2-10

We add all the numbers together, and all the variables

5X^2+2X-12

Back to the equation:

-(5X^2+2X-12)


We add all the numbers together, and all the variables

10X-(5X^2+2X-12)+4=0

We get rid of parentheses

-5X^2+10X-2X+12+4=0

We add all the numbers together, and all the variables

-5X^2+8X+16=0

a = -5; b = 8; c = +16;
Δ = b2-4ac
Δ = 82-4·(-5)·16
Δ = 384
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:

$\sqrt{\Delta}=\sqrt{384}=\sqrt{64*6}=\sqrt{64}*\sqrt{6}=8\sqrt{6}$
$X_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(8)-8\sqrt{6}}{2*-5}=\frac{-8-8\sqrt{6}}{-10}
$
$X_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(8)+8\sqrt{6}}{2*-5}=\frac{-8+8\sqrt{6}}{-10}
$