a^2+6a+7a=-5

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Solution for a^2+6a+7a=-5 equation:


Simplifying
a2 + 6a + 7a = -5

Reorder the terms:
6a + 7a + a2 = -5

Combine like terms: 6a + 7a = 13a
13a + a2 = -5

Solving
13a + a2 = -5

Solving for variable 'a'.

Reorder the terms:
5 + 13a + a2 = -5 + 5

Combine like terms: -5 + 5 = 0
5 + 13a + a2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-5' to each side of the equation.
5 + 13a + -5 + a2 = 0 + -5

Reorder the terms:
5 + -5 + 13a + a2 = 0 + -5

Combine like terms: 5 + -5 = 0
0 + 13a + a2 = 0 + -5
13a + a2 = 0 + -5

Combine like terms: 0 + -5 = -5
13a + a2 = -5

The a term is 13a.  Take half its coefficient (6.5).
Square it (42.25) and add it to both sides.

Add '42.25' to each side of the equation.
13a + 42.25 + a2 = -5 + 42.25

Reorder the terms:
42.25 + 13a + a2 = -5 + 42.25

Combine like terms: -5 + 42.25 = 37.25
42.25 + 13a + a2 = 37.25

Factor a perfect square on the left side:
(a + 6.5)(a + 6.5) = 37.25

Calculate the square root of the right side: 6.103277808

Break this problem into two subproblems by setting 
(a + 6.5) equal to 6.103277808 and -6.103277808.

Subproblem 1

a + 6.5 = 6.103277808 Simplifying a + 6.5 = 6.103277808 Reorder the terms: 6.5 + a = 6.103277808 Solving 6.5 + a = 6.103277808 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-6.5' to each side of the equation. 6.5 + -6.5 + a = 6.103277808 + -6.5 Combine like terms: 6.5 + -6.5 = 0.0 0.0 + a = 6.103277808 + -6.5 a = 6.103277808 + -6.5 Combine like terms: 6.103277808 + -6.5 = -0.396722192 a = -0.396722192 Simplifying a = -0.396722192

Subproblem 2

a + 6.5 = -6.103277808 Simplifying a + 6.5 = -6.103277808 Reorder the terms: 6.5 + a = -6.103277808 Solving 6.5 + a = -6.103277808 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-6.5' to each side of the equation. 6.5 + -6.5 + a = -6.103277808 + -6.5 Combine like terms: 6.5 + -6.5 = 0.0 0.0 + a = -6.103277808 + -6.5 a = -6.103277808 + -6.5 Combine like terms: -6.103277808 + -6.5 = -12.603277808 a = -12.603277808 Simplifying a = -12.603277808

Solution

The solution to the problem is based on the solutions from the subproblems. a = {-0.396722192, -12.603277808}

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