4b^2+23b=-23

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Solution for 4b^2+23b=-23 equation:


Simplifying
4b2 + 23b = -23

Reorder the terms:
23b + 4b2 = -23

Solving
23b + 4b2 = -23

Solving for variable 'b'.

Reorder the terms:
23 + 23b + 4b2 = -23 + 23

Combine like terms: -23 + 23 = 0
23 + 23b + 4b2 = 0

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

Divide each side by '4'.
5.75 + 5.75b + b2 = 0

Move the constant term to the right:

Add '-5.75' to each side of the equation.
5.75 + 5.75b + -5.75 + b2 = 0 + -5.75

Reorder the terms:
5.75 + -5.75 + 5.75b + b2 = 0 + -5.75

Combine like terms: 5.75 + -5.75 = 0.00
0.00 + 5.75b + b2 = 0 + -5.75
5.75b + b2 = 0 + -5.75

Combine like terms: 0 + -5.75 = -5.75
5.75b + b2 = -5.75

The b term is 5.75b.  Take half its coefficient (2.875).
Square it (8.265625) and add it to both sides.

Add '8.265625' to each side of the equation.
5.75b + 8.265625 + b2 = -5.75 + 8.265625

Reorder the terms:
8.265625 + 5.75b + b2 = -5.75 + 8.265625

Combine like terms: -5.75 + 8.265625 = 2.515625
8.265625 + 5.75b + b2 = 2.515625

Factor a perfect square on the left side:
(b + 2.875)(b + 2.875) = 2.515625

Calculate the square root of the right side: 1.586072193

Break this problem into two subproblems by setting 
(b + 2.875) equal to 1.586072193 and -1.586072193.

Subproblem 1

b + 2.875 = 1.586072193 Simplifying b + 2.875 = 1.586072193 Reorder the terms: 2.875 + b = 1.586072193 Solving 2.875 + b = 1.586072193 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-2.875' to each side of the equation. 2.875 + -2.875 + b = 1.586072193 + -2.875 Combine like terms: 2.875 + -2.875 = 0.000 0.000 + b = 1.586072193 + -2.875 b = 1.586072193 + -2.875 Combine like terms: 1.586072193 + -2.875 = -1.288927807 b = -1.288927807 Simplifying b = -1.288927807

Subproblem 2

b + 2.875 = -1.586072193 Simplifying b + 2.875 = -1.586072193 Reorder the terms: 2.875 + b = -1.586072193 Solving 2.875 + b = -1.586072193 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-2.875' to each side of the equation. 2.875 + -2.875 + b = -1.586072193 + -2.875 Combine like terms: 2.875 + -2.875 = 0.000 0.000 + b = -1.586072193 + -2.875 b = -1.586072193 + -2.875 Combine like terms: -1.586072193 + -2.875 = -4.461072193 b = -4.461072193 Simplifying b = -4.461072193

Solution

The solution to the problem is based on the solutions from the subproblems. b = {-1.288927807, -4.461072193}

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