2b^2=6b+7

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Solution for 2b^2=6b+7 equation:


Simplifying
2b2 = 6b + 7

Reorder the terms:
2b2 = 7 + 6b

Solving
2b2 = 7 + 6b

Solving for variable 'b'.

Reorder the terms:
-7 + -6b + 2b2 = 7 + 6b + -7 + -6b

Reorder the terms:
-7 + -6b + 2b2 = 7 + -7 + 6b + -6b

Combine like terms: 7 + -7 = 0
-7 + -6b + 2b2 = 0 + 6b + -6b
-7 + -6b + 2b2 = 6b + -6b

Combine like terms: 6b + -6b = 0
-7 + -6b + 2b2 = 0

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

Divide each side by '2'.
-3.5 + -3b + b2 = 0

Move the constant term to the right:

Add '3.5' to each side of the equation.
-3.5 + -3b + 3.5 + b2 = 0 + 3.5

Reorder the terms:
-3.5 + 3.5 + -3b + b2 = 0 + 3.5

Combine like terms: -3.5 + 3.5 = 0.0
0.0 + -3b + b2 = 0 + 3.5
-3b + b2 = 0 + 3.5

Combine like terms: 0 + 3.5 = 3.5
-3b + b2 = 3.5

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 = 3.5 + 2.25

Reorder the terms:
2.25 + -3b + b2 = 3.5 + 2.25

Combine like terms: 3.5 + 2.25 = 5.75
2.25 + -3b + b2 = 5.75

Factor a perfect square on the left side:
(b + -1.5)(b + -1.5) = 5.75

Calculate the square root of the right side: 2.397915762

Break this problem into two subproblems by setting 
(b + -1.5) equal to 2.397915762 and -2.397915762.

Subproblem 1

b + -1.5 = 2.397915762 Simplifying b + -1.5 = 2.397915762 Reorder the terms: -1.5 + b = 2.397915762 Solving -1.5 + b = 2.397915762 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 = 2.397915762 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + b = 2.397915762 + 1.5 b = 2.397915762 + 1.5 Combine like terms: 2.397915762 + 1.5 = 3.897915762 b = 3.897915762 Simplifying b = 3.897915762

Subproblem 2

b + -1.5 = -2.397915762 Simplifying b + -1.5 = -2.397915762 Reorder the terms: -1.5 + b = -2.397915762 Solving -1.5 + b = -2.397915762 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 = -2.397915762 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + b = -2.397915762 + 1.5 b = -2.397915762 + 1.5 Combine like terms: -2.397915762 + 1.5 = -0.897915762 b = -0.897915762 Simplifying b = -0.897915762

Solution

The solution to the problem is based on the solutions from the subproblems. b = {3.897915762, -0.897915762}

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