7a^2+53a+29=0

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Solution for 7a^2+53a+29=0 equation: 


Simplifying
7a2 + 53a + 29 = 0

Reorder the terms:
29 + 53a + 7a2 = 0

Solving
29 + 53a + 7a2 = 0

Solving for variable 'a'.

Begin completing the square.  Divide all terms by
7 the coefficient of the squared term: 

Divide each side by '7'.
4.142857143 + 7.571428571a + a2 = 0

Move the constant term to the right:

Add '-4.142857143' to each side of the equation.
4.142857143 + 7.571428571a + -4.142857143 + a2 = 0 + -4.142857143

Reorder the terms:
4.142857143 + -4.142857143 + 7.571428571a + a2 = 0 + -4.142857143

Combine like terms: 4.142857143 + -4.142857143 = 0.000000000
0.000000000 + 7.571428571a + a2 = 0 + -4.142857143
7.571428571a + a2 = 0 + -4.142857143

Combine like terms: 0 + -4.142857143 = -4.142857143
7.571428571a + a2 = -4.142857143

The a term is 7.571428571a.  Take half its coefficient (3.785714286).
Square it (14.33163266) and add it to both sides.

Add '14.33163266' to each side of the equation.
7.571428571a + 14.33163266 + a2 = -4.142857143 + 14.33163266

Reorder the terms:
14.33163266 + 7.571428571a + a2 = -4.142857143 + 14.33163266

Combine like terms: -4.142857143 + 14.33163266 = 10.188775517
14.33163266 + 7.571428571a + a2 = 10.188775517

Factor a perfect square on the left side:
(a + 3.785714286)(a + 3.785714286) = 10.188775517

Calculate the square root of the right side: 3.19198614

Break this problem into two subproblems by setting 
(a + 3.785714286) equal to 3.19198614 and -3.19198614.


Subproblem 1
a + 3.785714286 = 3.19198614

Simplifying
a + 3.785714286 = 3.19198614

Reorder the terms:
3.785714286 + a = 3.19198614

Solving
3.785714286 + a = 3.19198614

Solving for variable 'a'.

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

Add '-3.785714286' to each side of the equation.
3.785714286 + -3.785714286 + a = 3.19198614 + -3.785714286

Combine like terms: 3.785714286 + -3.785714286 = 0.000000000
0.000000000 + a = 3.19198614 + -3.785714286
a = 3.19198614 + -3.785714286

Combine like terms: 3.19198614 + -3.785714286 = -0.593728146
a = -0.593728146

Simplifying
a = -0.593728146


Subproblem 2
a + 3.785714286 = -3.19198614

Simplifying
a + 3.785714286 = -3.19198614

Reorder the terms:
3.785714286 + a = -3.19198614

Solving
3.785714286 + a = -3.19198614

Solving for variable 'a'.

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

Add '-3.785714286' to each side of the equation.
3.785714286 + -3.785714286 + a = -3.19198614 + -3.785714286

Combine like terms: 3.785714286 + -3.785714286 = 0.000000000
0.000000000 + a = -3.19198614 + -3.785714286
a = -3.19198614 + -3.785714286

Combine like terms: -3.19198614 + -3.785714286 = -6.977700426
a = -6.977700426

Simplifying
a = -6.977700426


Solution
The solution to the problem is based on the solutions
from the subproblems.
a = {-0.593728146, -6.977700426}

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