52b^2-156b-407=0

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Solution for 52b^2-156b-407=0 equation:


Simplifying
52b2 + -156b + -407 = 0

Reorder the terms:
-407 + -156b + 52b2 = 0

Solving
-407 + -156b + 52b2 = 0

Solving for variable 'b'.

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

Divide each side by '52'.
-7.826923077 + -3b + b2 = 0

Move the constant term to the right:

Add '7.826923077' to each side of the equation.
-7.826923077 + -3b + 7.826923077 + b2 = 0 + 7.826923077

Reorder the terms:
-7.826923077 + 7.826923077 + -3b + b2 = 0 + 7.826923077

Combine like terms: -7.826923077 + 7.826923077 = 0.000000000
0.000000000 + -3b + b2 = 0 + 7.826923077
-3b + b2 = 0 + 7.826923077

Combine like terms: 0 + 7.826923077 = 7.826923077
-3b + b2 = 7.826923077

The b term is -3b.  Take half its coefficient (-1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
-3b + 2.25 + b2 = 7.826923077 + 2.25

Reorder the terms:
2.25 + -3b + b2 = 7.826923077 + 2.25

Combine like terms: 7.826923077 + 2.25 = 10.076923077
2.25 + -3b + b2 = 10.076923077

Factor a perfect square on the left side:
(b + -1.5)(b + -1.5) = 10.076923077

Calculate the square root of the right side: 3.174416966

Break this problem into two subproblems by setting 
(b + -1.5) equal to 3.174416966 and -3.174416966.

Subproblem 1

b + -1.5 = 3.174416966 Simplifying b + -1.5 = 3.174416966 Reorder the terms: -1.5 + b = 3.174416966 Solving -1.5 + b = 3.174416966 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + b = 3.174416966 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + b = 3.174416966 + 1.5 b = 3.174416966 + 1.5 Combine like terms: 3.174416966 + 1.5 = 4.674416966 b = 4.674416966 Simplifying b = 4.674416966

Subproblem 2

b + -1.5 = -3.174416966 Simplifying b + -1.5 = -3.174416966 Reorder the terms: -1.5 + b = -3.174416966 Solving -1.5 + b = -3.174416966 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + b = -3.174416966 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + b = -3.174416966 + 1.5 b = -3.174416966 + 1.5 Combine like terms: -3.174416966 + 1.5 = -1.674416966 b = -1.674416966 Simplifying b = -1.674416966

Solution

The solution to the problem is based on the solutions from the subproblems. b = {4.674416966, -1.674416966}

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