4a^2+28a^2+a+7=0

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Solution for 4a^2+28a^2+a+7=0 equation:


Simplifying
4a2 + 28a2 + a + 7 = 0

Reorder the terms:
7 + a + 4a2 + 28a2 = 0

Combine like terms: 4a2 + 28a2 = 32a2
7 + a + 32a2 = 0

Solving
7 + a + 32a2 = 0

Solving for variable 'a'.

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

Divide each side by '32'.
0.21875 + 0.03125a + a2 = 0

Move the constant term to the right:

Add '-0.21875' to each side of the equation.
0.21875 + 0.03125a + -0.21875 + a2 = 0 + -0.21875

Reorder the terms:
0.21875 + -0.21875 + 0.03125a + a2 = 0 + -0.21875

Combine like terms: 0.21875 + -0.21875 = 0.00000
0.00000 + 0.03125a + a2 = 0 + -0.21875
0.03125a + a2 = 0 + -0.21875

Combine like terms: 0 + -0.21875 = -0.21875
0.03125a + a2 = -0.21875

The a term is a.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
0.03125a + 0.25 + a2 = -0.21875 + 0.25

Reorder the terms:
0.25 + 0.03125a + a2 = -0.21875 + 0.25

Combine like terms: -0.21875 + 0.25 = 0.03125
0.25 + 0.03125a + a2 = 0.03125

Factor a perfect square on the left side:
(a + 0.5)(a + 0.5) = 0.03125

Calculate the square root of the right side: 0.176776695

Break this problem into two subproblems by setting 
(a + 0.5) equal to 0.176776695 and -0.176776695.

Subproblem 1

a + 0.5 = 0.176776695 Simplifying a + 0.5 = 0.176776695 Reorder the terms: 0.5 + a = 0.176776695 Solving 0.5 + a = 0.176776695 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + a = 0.176776695 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + a = 0.176776695 + -0.5 a = 0.176776695 + -0.5 Combine like terms: 0.176776695 + -0.5 = -0.323223305 a = -0.323223305 Simplifying a = -0.323223305

Subproblem 2

a + 0.5 = -0.176776695 Simplifying a + 0.5 = -0.176776695 Reorder the terms: 0.5 + a = -0.176776695 Solving 0.5 + a = -0.176776695 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + a = -0.176776695 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + a = -0.176776695 + -0.5 a = -0.176776695 + -0.5 Combine like terms: -0.176776695 + -0.5 = -0.676776695 a = -0.676776695 Simplifying a = -0.676776695

Solution

The solution to the problem is based on the solutions from the subproblems. a = {-0.323223305, -0.676776695}

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