7a^2=7-40a

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


Simplifying
7a2 = 7 + -40a

Solving
7a2 = 7 + -40a

Solving for variable 'a'.

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

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

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

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

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

Divide each side by '7'.
-1 + 5.714285714a + a2 = 0

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + 5.714285714a + 1 + a2 = 0 + 1

Reorder the terms:
-1 + 1 + 5.714285714a + a2 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + 5.714285714a + a2 = 0 + 1
5.714285714a + a2 = 0 + 1

Combine like terms: 0 + 1 = 1
5.714285714a + a2 = 1

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 = 1 + 8.163265305

Reorder the terms:
8.163265305 + 5.714285714a + a2 = 1 + 8.163265305

Combine like terms: 1 + 8.163265305 = 9.163265305
8.163265305 + 5.714285714a + a2 = 9.163265305

Factor a perfect square on the left side:
(a + 2.857142857)(a + 2.857142857) = 9.163265305

Calculate the square root of the right side: 3.027088586

Break this problem into two subproblems by setting 
(a + 2.857142857) equal to 3.027088586 and -3.027088586.

Subproblem 1

a + 2.857142857 = 3.027088586 Simplifying a + 2.857142857 = 3.027088586 Reorder the terms: 2.857142857 + a = 3.027088586 Solving 2.857142857 + a = 3.027088586 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 = 3.027088586 + -2.857142857 Combine like terms: 2.857142857 + -2.857142857 = 0.000000000 0.000000000 + a = 3.027088586 + -2.857142857 a = 3.027088586 + -2.857142857 Combine like terms: 3.027088586 + -2.857142857 = 0.169945729 a = 0.169945729 Simplifying a = 0.169945729

Subproblem 2

a + 2.857142857 = -3.027088586 Simplifying a + 2.857142857 = -3.027088586 Reorder the terms: 2.857142857 + a = -3.027088586 Solving 2.857142857 + a = -3.027088586 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 = -3.027088586 + -2.857142857 Combine like terms: 2.857142857 + -2.857142857 = 0.000000000 0.000000000 + a = -3.027088586 + -2.857142857 a = -3.027088586 + -2.857142857 Combine like terms: -3.027088586 + -2.857142857 = -5.884231443 a = -5.884231443 Simplifying a = -5.884231443

Solution

The solution to the problem is based on the solutions from the subproblems. a = {0.169945729, -5.884231443}

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