Too many terms!

Simplify the equation:

$a^{2}(a^{2} + 1) = 2a^{3}$

$a^{2}(a^{2} + 1) - 2a^{3} = 0$

$a^{2}(a^{2} + 1 - 2a) = 0$

$a = 0$ or $a^{2} - 2a + 1 = 0$

Which gives a = 0, $(a-1)^{2} = 0$

$a = 0, a = 1$

Since $x = y + 1$ , we have $x = 1$ and $y = 0$

Therefore, $18 \times |x + y| = 18 \times |1 + 0| = 18$

That's the answer!

By expanding and shifting the terms over to the LHS, we get

a^4-2a^3+a^2=0

Taking out common factor a^2 , we get

a^2(a-1)^2=0

a= 0 or 1

Therefore,

18 x |1+0| = 18

The equation has too many terms indeed! This equation can actually be simplified as a^2-2a+1=0. And it is clear that the only solution for the equation is x=1. Since y=x-1, then y=0. Now we have x=1 and y=0. Adding those two value gives x+y=1, and multiply this with 18 gives 18 too.

a^2(a^2+1)+18 =2a^3+18 can be simplified to a^2(a^2+1)=2a^3. Division by a^2 is allowed only if a is not equal to zero. a=0 satisfies this equation and is one of the solutions. For a's not equal to zero, we can proceed with a^2-2a^3+1=0. This gives the solution a=1(double root). x=1 and y=0 which satisfies x=y+1. 18(0+1)=18.

a^2 x (a^2 + 1) + 18 = 2 a^3 + 18 <=> a^2 x (a-1)^2 <=> a= 0 or a = 1. x = y + 1 => x = 1, y = 0 <=> 18 x |x + y | = 18.