7a^2+32=-40a

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Solution for 7a^2+32=-40a equation:


Simplifying
7a2 + 32 = -40a

Reorder the terms:
32 + 7a2 = -40a

Solving
32 + 7a2 = -40a

Solving for variable 'a'.

Reorder the terms:
32 + 40a + 7a2 = -40a + 40a

Combine like terms: -40a + 40a = 0
32 + 40a + 7a2 = 0

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

Divide each side by '7'.
4.571428571 + 5.714285714a + a2 = 0

Move the constant term to the right:

Add '-4.571428571' to each side of the equation.
4.571428571 + 5.714285714a + -4.571428571 + a2 = 0 + -4.571428571

Reorder the terms:
4.571428571 + -4.571428571 + 5.714285714a + a2 = 0 + -4.571428571

Combine like terms: 4.571428571 + -4.571428571 = 0.000000000
0.000000000 + 5.714285714a + a2 = 0 + -4.571428571
5.714285714a + a2 = 0 + -4.571428571

Combine like terms: 0 + -4.571428571 = -4.571428571
5.714285714a + a2 = -4.571428571

The a term is 5.714285714a.  Take half its coefficient (2.857142857).
Square it (8.163265305) and add it to both sides.

Add '8.163265305' to each side of the equation.
5.714285714a + 8.163265305 + a2 = -4.571428571 + 8.163265305

Reorder the terms:
8.163265305 + 5.714285714a + a2 = -4.571428571 + 8.163265305

Combine like terms: -4.571428571 + 8.163265305 = 3.591836734
8.163265305 + 5.714285714a + a2 = 3.591836734

Factor a perfect square on the left side:
(a + 2.857142857)(a + 2.857142857) = 3.591836734

Calculate the square root of the right side: 1.895214166

Break this problem into two subproblems by setting 
(a + 2.857142857) equal to 1.895214166 and -1.895214166.

Subproblem 1

a + 2.857142857 = 1.895214166 Simplifying a + 2.857142857 = 1.895214166 Reorder the terms: 2.857142857 + a = 1.895214166 Solving 2.857142857 + a = 1.895214166 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-2.857142857' to each side of the equation. 2.857142857 + -2.857142857 + a = 1.895214166 + -2.857142857 Combine like terms: 2.857142857 + -2.857142857 = 0.000000000 0.000000000 + a = 1.895214166 + -2.857142857 a = 1.895214166 + -2.857142857 Combine like terms: 1.895214166 + -2.857142857 = -0.961928691 a = -0.961928691 Simplifying a = -0.961928691

Subproblem 2

a + 2.857142857 = -1.895214166 Simplifying a + 2.857142857 = -1.895214166 Reorder the terms: 2.857142857 + a = -1.895214166 Solving 2.857142857 + a = -1.895214166 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-2.857142857' to each side of the equation. 2.857142857 + -2.857142857 + a = -1.895214166 + -2.857142857 Combine like terms: 2.857142857 + -2.857142857 = 0.000000000 0.000000000 + a = -1.895214166 + -2.857142857 a = -1.895214166 + -2.857142857 Combine like terms: -1.895214166 + -2.857142857 = -4.752357023 a = -4.752357023 Simplifying a = -4.752357023

Solution

The solution to the problem is based on the solutions from the subproblems. a = {-0.961928691, -4.752357023}

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