4b^2+b-19=0

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Solution for 4b^2+b-19=0 equation:


Simplifying
4b2 + b + -19 = 0

Reorder the terms:
-19 + b + 4b2 = 0

Solving
-19 + b + 4b2 = 0

Solving for variable 'b'.

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

Divide each side by '4'.
-4.75 + 0.25b + b2 = 0

Move the constant term to the right:

Add '4.75' to each side of the equation.
-4.75 + 0.25b + 4.75 + b2 = 0 + 4.75

Reorder the terms:
-4.75 + 4.75 + 0.25b + b2 = 0 + 4.75

Combine like terms: -4.75 + 4.75 = 0.00
0.00 + 0.25b + b2 = 0 + 4.75
0.25b + b2 = 0 + 4.75

Combine like terms: 0 + 4.75 = 4.75
0.25b + b2 = 4.75

The b term is b.  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.
0.25b + 0.25 + b2 = 4.75 + 0.25

Reorder the terms:
0.25 + 0.25b + b2 = 4.75 + 0.25

Combine like terms: 4.75 + 0.25 = 5
0.25 + 0.25b + b2 = 5

Factor a perfect square on the left side:
(b + 0.5)(b + 0.5) = 5

Calculate the square root of the right side: 2.236067978

Break this problem into two subproblems by setting 
(b + 0.5) equal to 2.236067978 and -2.236067978.

Subproblem 1

b + 0.5 = 2.236067978 Simplifying b + 0.5 = 2.236067978 Reorder the terms: 0.5 + b = 2.236067978 Solving 0.5 + b = 2.236067978 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + b = 2.236067978 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + b = 2.236067978 + -0.5 b = 2.236067978 + -0.5 Combine like terms: 2.236067978 + -0.5 = 1.736067978 b = 1.736067978 Simplifying b = 1.736067978

Subproblem 2

b + 0.5 = -2.236067978 Simplifying b + 0.5 = -2.236067978 Reorder the terms: 0.5 + b = -2.236067978 Solving 0.5 + b = -2.236067978 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + b = -2.236067978 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + b = -2.236067978 + -0.5 b = -2.236067978 + -0.5 Combine like terms: -2.236067978 + -0.5 = -2.736067978 b = -2.736067978 Simplifying b = -2.736067978

Solution

The solution to the problem is based on the solutions from the subproblems. b = {1.736067978, -2.736067978}

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