3b^2+7b-3=0

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Solution for 3b^2+7b-3=0 equation:


Simplifying
3b2 + 7b + -3 = 0

Reorder the terms:
-3 + 7b + 3b2 = 0

Solving
-3 + 7b + 3b2 = 0

Solving for variable 'b'.

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

Divide each side by '3'.
-1 + 2.333333333b + b2 = 0

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + 2.333333333b + 1 + b2 = 0 + 1

Reorder the terms:
-1 + 1 + 2.333333333b + b2 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + 2.333333333b + b2 = 0 + 1
2.333333333b + b2 = 0 + 1

Combine like terms: 0 + 1 = 1
2.333333333b + b2 = 1

The b term is 2.333333333b.  Take half its coefficient (1.166666667).
Square it (1.361111112) and add it to both sides.

Add '1.361111112' to each side of the equation.
2.333333333b + 1.361111112 + b2 = 1 + 1.361111112

Reorder the terms:
1.361111112 + 2.333333333b + b2 = 1 + 1.361111112

Combine like terms: 1 + 1.361111112 = 2.361111112
1.361111112 + 2.333333333b + b2 = 2.361111112

Factor a perfect square on the left side:
(b + 1.166666667)(b + 1.166666667) = 2.361111112

Calculate the square root of the right side: 1.536590743

Break this problem into two subproblems by setting 
(b + 1.166666667) equal to 1.536590743 and -1.536590743.

Subproblem 1

b + 1.166666667 = 1.536590743 Simplifying b + 1.166666667 = 1.536590743 Reorder the terms: 1.166666667 + b = 1.536590743 Solving 1.166666667 + b = 1.536590743 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-1.166666667' to each side of the equation. 1.166666667 + -1.166666667 + b = 1.536590743 + -1.166666667 Combine like terms: 1.166666667 + -1.166666667 = 0.000000000 0.000000000 + b = 1.536590743 + -1.166666667 b = 1.536590743 + -1.166666667 Combine like terms: 1.536590743 + -1.166666667 = 0.369924076 b = 0.369924076 Simplifying b = 0.369924076

Subproblem 2

b + 1.166666667 = -1.536590743 Simplifying b + 1.166666667 = -1.536590743 Reorder the terms: 1.166666667 + b = -1.536590743 Solving 1.166666667 + b = -1.536590743 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-1.166666667' to each side of the equation. 1.166666667 + -1.166666667 + b = -1.536590743 + -1.166666667 Combine like terms: 1.166666667 + -1.166666667 = 0.000000000 0.000000000 + b = -1.536590743 + -1.166666667 b = -1.536590743 + -1.166666667 Combine like terms: -1.536590743 + -1.166666667 = -2.70325741 b = -2.70325741 Simplifying b = -2.70325741

Solution

The solution to the problem is based on the solutions from the subproblems. b = {0.369924076, -2.70325741}

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