2a+1=-a(a+5)

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Solution for 2a+1=-a(a+5) equation:


Simplifying
2a + 1 = -1a(a + 5)

Reorder the terms:
1 + 2a = -1a(a + 5)

Reorder the terms:
1 + 2a = -1a(5 + a)
1 + 2a = (5 * -1a + a * -1a)
1 + 2a = (-5a + -1a2)

Solving
1 + 2a = -5a + -1a2

Solving for variable 'a'.

Combine like terms: 2a + 5a = 7a
1 + 7a + a2 = -5a + -1a2 + 5a + a2

Reorder the terms:
1 + 7a + a2 = -5a + 5a + -1a2 + a2

Combine like terms: -5a + 5a = 0
1 + 7a + a2 = 0 + -1a2 + a2
1 + 7a + a2 = -1a2 + a2

Combine like terms: -1a2 + a2 = 0
1 + 7a + a2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-1' to each side of the equation.
1 + 7a + -1 + a2 = 0 + -1

Reorder the terms:
1 + -1 + 7a + a2 = 0 + -1

Combine like terms: 1 + -1 = 0
0 + 7a + a2 = 0 + -1
7a + a2 = 0 + -1

Combine like terms: 0 + -1 = -1
7a + a2 = -1

The a term is 7a.  Take half its coefficient (3.5).
Square it (12.25) and add it to both sides.

Add '12.25' to each side of the equation.
7a + 12.25 + a2 = -1 + 12.25

Reorder the terms:
12.25 + 7a + a2 = -1 + 12.25

Combine like terms: -1 + 12.25 = 11.25
12.25 + 7a + a2 = 11.25

Factor a perfect square on the left side:
(a + 3.5)(a + 3.5) = 11.25

Calculate the square root of the right side: 3.354101966

Break this problem into two subproblems by setting 
(a + 3.5) equal to 3.354101966 and -3.354101966.

Subproblem 1

a + 3.5 = 3.354101966 Simplifying a + 3.5 = 3.354101966 Reorder the terms: 3.5 + a = 3.354101966 Solving 3.5 + a = 3.354101966 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-3.5' to each side of the equation. 3.5 + -3.5 + a = 3.354101966 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + a = 3.354101966 + -3.5 a = 3.354101966 + -3.5 Combine like terms: 3.354101966 + -3.5 = -0.145898034 a = -0.145898034 Simplifying a = -0.145898034

Subproblem 2

a + 3.5 = -3.354101966 Simplifying a + 3.5 = -3.354101966 Reorder the terms: 3.5 + a = -3.354101966 Solving 3.5 + a = -3.354101966 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-3.5' to each side of the equation. 3.5 + -3.5 + a = -3.354101966 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + a = -3.354101966 + -3.5 a = -3.354101966 + -3.5 Combine like terms: -3.354101966 + -3.5 = -6.854101966 a = -6.854101966 Simplifying a = -6.854101966

Solution

The solution to the problem is based on the solutions from the subproblems. a = {-0.145898034, -6.854101966}

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