6b^2+16b-9=0

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Solution for 6b^2+16b-9=0 equation:


Simplifying
6b2 + 16b + -9 = 0

Reorder the terms:
-9 + 16b + 6b2 = 0

Solving
-9 + 16b + 6b2 = 0

Solving for variable 'b'.

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

Divide each side by '6'.
-1.5 + 2.666666667b + b2 = 0

Move the constant term to the right:

Add '1.5' to each side of the equation.
-1.5 + 2.666666667b + 1.5 + b2 = 0 + 1.5

Reorder the terms:
-1.5 + 1.5 + 2.666666667b + b2 = 0 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + 2.666666667b + b2 = 0 + 1.5
2.666666667b + b2 = 0 + 1.5

Combine like terms: 0 + 1.5 = 1.5
2.666666667b + b2 = 1.5

The b term is 2.666666667b.  Take half its coefficient (1.333333334).
Square it (1.777777780) and add it to both sides.

Add '1.777777780' to each side of the equation.
2.666666667b + 1.777777780 + b2 = 1.5 + 1.777777780

Reorder the terms:
1.777777780 + 2.666666667b + b2 = 1.5 + 1.777777780

Combine like terms: 1.5 + 1.777777780 = 3.27777778
1.777777780 + 2.666666667b + b2 = 3.27777778

Factor a perfect square on the left side:
(b + 1.333333334)(b + 1.333333334) = 3.27777778

Calculate the square root of the right side: 1.810463416

Break this problem into two subproblems by setting 
(b + 1.333333334) equal to 1.810463416 and -1.810463416.

Subproblem 1

b + 1.333333334 = 1.810463416 Simplifying b + 1.333333334 = 1.810463416 Reorder the terms: 1.333333334 + b = 1.810463416 Solving 1.333333334 + b = 1.810463416 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-1.333333334' to each side of the equation. 1.333333334 + -1.333333334 + b = 1.810463416 + -1.333333334 Combine like terms: 1.333333334 + -1.333333334 = 0.000000000 0.000000000 + b = 1.810463416 + -1.333333334 b = 1.810463416 + -1.333333334 Combine like terms: 1.810463416 + -1.333333334 = 0.477130082 b = 0.477130082 Simplifying b = 0.477130082

Subproblem 2

b + 1.333333334 = -1.810463416 Simplifying b + 1.333333334 = -1.810463416 Reorder the terms: 1.333333334 + b = -1.810463416 Solving 1.333333334 + b = -1.810463416 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-1.333333334' to each side of the equation. 1.333333334 + -1.333333334 + b = -1.810463416 + -1.333333334 Combine like terms: 1.333333334 + -1.333333334 = 0.000000000 0.000000000 + b = -1.810463416 + -1.333333334 b = -1.810463416 + -1.333333334 Combine like terms: -1.810463416 + -1.333333334 = -3.14379675 b = -3.14379675 Simplifying b = -3.14379675

Solution

The solution to the problem is based on the solutions from the subproblems. b = {0.477130082, -3.14379675}

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