6b-b^2=7

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


Simplifying
6b + -1b2 = 7

Solving
6b + -1b2 = 7

Solving for variable 'b'.

Reorder the terms:
-7 + 6b + -1b2 = 7 + -7

Combine like terms: 7 + -7 = 0
-7 + 6b + -1b2 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
7 + -6b + b2 = 0

Move the constant term to the right:

Add '-7' to each side of the equation.
7 + -6b + -7 + b2 = 0 + -7

Reorder the terms:
7 + -7 + -6b + b2 = 0 + -7

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

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

The b term is -6b.  Take half its coefficient (-3).
Square it (9) and add it to both sides.

Add '9' to each side of the equation.
-6b + 9 + b2 = -7 + 9

Reorder the terms:
9 + -6b + b2 = -7 + 9

Combine like terms: -7 + 9 = 2
9 + -6b + b2 = 2

Factor a perfect square on the left side:
(b + -3)(b + -3) = 2

Calculate the square root of the right side: 1.414213562

Break this problem into two subproblems by setting 
(b + -3) equal to 1.414213562 and -1.414213562.

Subproblem 1

b + -3 = 1.414213562 Simplifying b + -3 = 1.414213562 Reorder the terms: -3 + b = 1.414213562 Solving -3 + b = 1.414213562 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + b = 1.414213562 + 3 Combine like terms: -3 + 3 = 0 0 + b = 1.414213562 + 3 b = 1.414213562 + 3 Combine like terms: 1.414213562 + 3 = 4.414213562 b = 4.414213562 Simplifying b = 4.414213562

Subproblem 2

b + -3 = -1.414213562 Simplifying b + -3 = -1.414213562 Reorder the terms: -3 + b = -1.414213562 Solving -3 + b = -1.414213562 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + b = -1.414213562 + 3 Combine like terms: -3 + 3 = 0 0 + b = -1.414213562 + 3 b = -1.414213562 + 3 Combine like terms: -1.414213562 + 3 = 1.585786438 b = 1.585786438 Simplifying b = 1.585786438

Solution

The solution to the problem is based on the solutions from the subproblems. b = {4.414213562, 1.585786438}

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