a^2+2a=32223

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Solution for a^2+2a=32223 equation:


Simplifying
a2 + 2a = 32223

Reorder the terms:
2a + a2 = 32223

Solving
2a + a2 = 32223

Solving for variable 'a'.

Reorder the terms:
-32223 + 2a + a2 = 32223 + -32223

Combine like terms: 32223 + -32223 = 0
-32223 + 2a + a2 = 0

Begin completing the square.

Move the constant term to the right:

Add '32223' to each side of the equation.
-32223 + 2a + 32223 + a2 = 0 + 32223

Reorder the terms:
-32223 + 32223 + 2a + a2 = 0 + 32223

Combine like terms: -32223 + 32223 = 0
0 + 2a + a2 = 0 + 32223
2a + a2 = 0 + 32223

Combine like terms: 0 + 32223 = 32223
2a + a2 = 32223

The a term is 2a.  Take half its coefficient (1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
2a + 1 + a2 = 32223 + 1

Reorder the terms:
1 + 2a + a2 = 32223 + 1

Combine like terms: 32223 + 1 = 32224
1 + 2a + a2 = 32224

Factor a perfect square on the left side:
(a + 1)(a + 1) = 32224

Calculate the square root of the right side: 179.510445379

Break this problem into two subproblems by setting 
(a + 1) equal to 179.510445379 and -179.510445379.

Subproblem 1

a + 1 = 179.510445379 Simplifying a + 1 = 179.510445379 Reorder the terms: 1 + a = 179.510445379 Solving 1 + a = 179.510445379 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + a = 179.510445379 + -1 Combine like terms: 1 + -1 = 0 0 + a = 179.510445379 + -1 a = 179.510445379 + -1 Combine like terms: 179.510445379 + -1 = 178.510445379 a = 178.510445379 Simplifying a = 178.510445379

Subproblem 2

a + 1 = -179.510445379 Simplifying a + 1 = -179.510445379 Reorder the terms: 1 + a = -179.510445379 Solving 1 + a = -179.510445379 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + a = -179.510445379 + -1 Combine like terms: 1 + -1 = 0 0 + a = -179.510445379 + -1 a = -179.510445379 + -1 Combine like terms: -179.510445379 + -1 = -180.510445379 a = -180.510445379 Simplifying a = -180.510445379

Solution

The solution to the problem is based on the solutions from the subproblems. a = {178.510445379, -180.510445379}

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