a^2+12a+23=7

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Solution for a^2+12a+23=7 equation:


Simplifying
a2 + 12a + 23 = 7

Reorder the terms:
23 + 12a + a2 = 7

Solving
23 + 12a + a2 = 7

Solving for variable 'a'.

Reorder the terms:
23 + -7 + 12a + a2 = 7 + -7

Combine like terms: 23 + -7 = 16
16 + 12a + a2 = 7 + -7

Combine like terms: 7 + -7 = 0
16 + 12a + a2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-16' to each side of the equation.
16 + 12a + -16 + a2 = 0 + -16

Reorder the terms:
16 + -16 + 12a + a2 = 0 + -16

Combine like terms: 16 + -16 = 0
0 + 12a + a2 = 0 + -16
12a + a2 = 0 + -16

Combine like terms: 0 + -16 = -16
12a + a2 = -16

The a term is 12a.  Take half its coefficient (6).
Square it (36) and add it to both sides.

Add '36' to each side of the equation.
12a + 36 + a2 = -16 + 36

Reorder the terms:
36 + 12a + a2 = -16 + 36

Combine like terms: -16 + 36 = 20
36 + 12a + a2 = 20

Factor a perfect square on the left side:
(a + 6)(a + 6) = 20

Calculate the square root of the right side: 4.472135955

Break this problem into two subproblems by setting 
(a + 6) equal to 4.472135955 and -4.472135955.

Subproblem 1

a + 6 = 4.472135955 Simplifying a + 6 = 4.472135955 Reorder the terms: 6 + a = 4.472135955 Solving 6 + a = 4.472135955 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-6' to each side of the equation. 6 + -6 + a = 4.472135955 + -6 Combine like terms: 6 + -6 = 0 0 + a = 4.472135955 + -6 a = 4.472135955 + -6 Combine like terms: 4.472135955 + -6 = -1.527864045 a = -1.527864045 Simplifying a = -1.527864045

Subproblem 2

a + 6 = -4.472135955 Simplifying a + 6 = -4.472135955 Reorder the terms: 6 + a = -4.472135955 Solving 6 + a = -4.472135955 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-6' to each side of the equation. 6 + -6 + a = -4.472135955 + -6 Combine like terms: 6 + -6 = 0 0 + a = -4.472135955 + -6 a = -4.472135955 + -6 Combine like terms: -4.472135955 + -6 = -10.472135955 a = -10.472135955 Simplifying a = -10.472135955

Solution

The solution to the problem is based on the solutions from the subproblems. a = {-1.527864045, -10.472135955}

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