a+(a+1)*(a+1)=505

Solution for a+(a+1)*(a+1)=505 equation:

Simplifying
a + (a + 1)(a + 1) = 505

Reorder the terms:
a + (1 + a)(a + 1) = 505

Reorder the terms:
a + (1 + a)(1 + a) = 505

Multiply (1 + a) * (1 + a)
a + (1(1 + a) + a(1 + a)) = 505
a + ((1 * 1 + a * 1) + a(1 + a)) = 505
a + ((1 + 1a) + a(1 + a)) = 505
a + (1 + 1a + (1 * a + a * a)) = 505
a + (1 + 1a + (1a + a2)) = 505

Combine like terms: 1a + 1a = 2a
a + (1 + 2a + a2) = 505

Reorder the terms:
1 + a + 2a + a2 = 505

Combine like terms: a + 2a = 3a
1 + 3a + a2 = 505

Solving
1 + 3a + a2 = 505

Solving for variable 'a'.

Reorder the terms:
1 + -505 + 3a + a2 = 505 + -505

Combine like terms: 1 + -505 = -504
-504 + 3a + a2 = 505 + -505

Combine like terms: 505 + -505 = 0
-504 + 3a + a2 = 0

Factor a trinomial.
(-24 + -1a)(21 + -1a) = 0

Subproblem 1
Set the factor '(-24 + -1a)' equal to zero and attempt to solve:

Simplifying
-24 + -1a = 0

Solving
-24 + -1a = 0

Move all terms containing a to the left, all other terms to the right.

Add '24' to each side of the equation.
-24 + 24 + -1a = 0 + 24

Combine like terms: -24 + 24 = 0
0 + -1a = 0 + 24
-1a = 0 + 24

Combine like terms: 0 + 24 = 24
-1a = 24

Divide each side by '-1'.
a = -24

Simplifying
a = -24
Subproblem 2
Set the factor '(21 + -1a)' equal to zero and attempt to solve:

Simplifying
21 + -1a = 0

Solving
21 + -1a = 0

Move all terms containing a to the left, all other terms to the right.

Add '-21' to each side of the equation.
21 + -21 + -1a = 0 + -21

Combine like terms: 21 + -21 = 0
0 + -1a = 0 + -21
-1a = 0 + -21

Combine like terms: 0 + -21 = -21
-1a = -21

Divide each side by '-1'.
a = 21

Simplifying
a = 21
Solution
a = {-24, 21}