7a^2+7+32a=0

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Solution for 7a^2+7+32a=0 equation:


Simplifying
7a2 + 7 + 32a = 0

Reorder the terms:
7 + 32a + 7a2 = 0

Solving
7 + 32a + 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'.
1 + 4.571428571a + a2 = 0

Move the constant term to the right:

Add '-1' to each side of the equation.
1 + 4.571428571a + -1 + a2 = 0 + -1

Reorder the terms:
1 + -1 + 4.571428571a + a2 = 0 + -1

Combine like terms: 1 + -1 = 0
0 + 4.571428571a + a2 = 0 + -1
4.571428571a + a2 = 0 + -1

Combine like terms: 0 + -1 = -1
4.571428571a + a2 = -1

The a term is 4.571428571a.  Take half its coefficient (2.285714286).
Square it (5.224489797) and add it to both sides.

Add '5.224489797' to each side of the equation.
4.571428571a + 5.224489797 + a2 = -1 + 5.224489797

Reorder the terms:
5.224489797 + 4.571428571a + a2 = -1 + 5.224489797

Combine like terms: -1 + 5.224489797 = 4.224489797
5.224489797 + 4.571428571a + a2 = 4.224489797

Factor a perfect square on the left side:
(a + 2.285714286)(a + 2.285714286) = 4.224489797

Calculate the square root of the right side: 2.055356367

Break this problem into two subproblems by setting 
(a + 2.285714286) equal to 2.055356367 and -2.055356367.

Subproblem 1

a + 2.285714286 = 2.055356367 Simplifying a + 2.285714286 = 2.055356367 Reorder the terms: 2.285714286 + a = 2.055356367 Solving 2.285714286 + a = 2.055356367 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-2.285714286' to each side of the equation. 2.285714286 + -2.285714286 + a = 2.055356367 + -2.285714286 Combine like terms: 2.285714286 + -2.285714286 = 0.000000000 0.000000000 + a = 2.055356367 + -2.285714286 a = 2.055356367 + -2.285714286 Combine like terms: 2.055356367 + -2.285714286 = -0.230357919 a = -0.230357919 Simplifying a = -0.230357919

Subproblem 2

a + 2.285714286 = -2.055356367 Simplifying a + 2.285714286 = -2.055356367 Reorder the terms: 2.285714286 + a = -2.055356367 Solving 2.285714286 + a = -2.055356367 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-2.285714286' to each side of the equation. 2.285714286 + -2.285714286 + a = -2.055356367 + -2.285714286 Combine like terms: 2.285714286 + -2.285714286 = 0.000000000 0.000000000 + a = -2.055356367 + -2.285714286 a = -2.055356367 + -2.285714286 Combine like terms: -2.055356367 + -2.285714286 = -4.341070653 a = -4.341070653 Simplifying a = -4.341070653

Solution

The solution to the problem is based on the solutions from the subproblems. a = {-0.230357919, -4.341070653}

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