a^2+6a=-3

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Solution for a^2+6a=-3 equation:


Simplifying
a2 + 6a = -3

Reorder the terms:
6a + a2 = -3

Solving
6a + a2 = -3

Solving for variable 'a'.

Reorder the terms:
3 + 6a + a2 = -3 + 3

Combine like terms: -3 + 3 = 0
3 + 6a + a2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-3' to each side of the equation.
3 + 6a + -3 + a2 = 0 + -3

Reorder the terms:
3 + -3 + 6a + a2 = 0 + -3

Combine like terms: 3 + -3 = 0
0 + 6a + a2 = 0 + -3
6a + a2 = 0 + -3

Combine like terms: 0 + -3 = -3
6a + a2 = -3

The a term is 6a.  Take half its coefficient (3).
Square it (9) and add it to both sides.

Add '9' to each side of the equation.
6a + 9 + a2 = -3 + 9

Reorder the terms:
9 + 6a + a2 = -3 + 9

Combine like terms: -3 + 9 = 6
9 + 6a + a2 = 6

Factor a perfect square on the left side:
(a + 3)(a + 3) = 6

Calculate the square root of the right side: 2.449489743

Break this problem into two subproblems by setting 
(a + 3) equal to 2.449489743 and -2.449489743.

Subproblem 1

a + 3 = 2.449489743 Simplifying a + 3 = 2.449489743 Reorder the terms: 3 + a = 2.449489743 Solving 3 + a = 2.449489743 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + a = 2.449489743 + -3 Combine like terms: 3 + -3 = 0 0 + a = 2.449489743 + -3 a = 2.449489743 + -3 Combine like terms: 2.449489743 + -3 = -0.550510257 a = -0.550510257 Simplifying a = -0.550510257

Subproblem 2

a + 3 = -2.449489743 Simplifying a + 3 = -2.449489743 Reorder the terms: 3 + a = -2.449489743 Solving 3 + a = -2.449489743 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + a = -2.449489743 + -3 Combine like terms: 3 + -3 = 0 0 + a = -2.449489743 + -3 a = -2.449489743 + -3 Combine like terms: -2.449489743 + -3 = -5.449489743 a = -5.449489743 Simplifying a = -5.449489743

Solution

The solution to the problem is based on the solutions from the subproblems. a = {-0.550510257, -5.449489743}

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