b^2-7b=23

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Solution for b^2-7b=23 equation:


Simplifying
b2 + -7b = 23

Reorder the terms:
-7b + b2 = 23

Solving
-7b + b2 = 23

Solving for variable 'b'.

Reorder the terms:
-23 + -7b + b2 = 23 + -23

Combine like terms: 23 + -23 = 0
-23 + -7b + b2 = 0

Begin completing the square.

Move the constant term to the right:

Add '23' to each side of the equation.
-23 + -7b + 23 + b2 = 0 + 23

Reorder the terms:
-23 + 23 + -7b + b2 = 0 + 23

Combine like terms: -23 + 23 = 0
0 + -7b + b2 = 0 + 23
-7b + b2 = 0 + 23

Combine like terms: 0 + 23 = 23
-7b + b2 = 23

The b term is -7b.  Take half its coefficient (-3.5).
Square it (12.25) and add it to both sides.

Add '12.25' to each side of the equation.
-7b + 12.25 + b2 = 23 + 12.25

Reorder the terms:
12.25 + -7b + b2 = 23 + 12.25

Combine like terms: 23 + 12.25 = 35.25
12.25 + -7b + b2 = 35.25

Factor a perfect square on the left side:
(b + -3.5)(b + -3.5) = 35.25

Calculate the square root of the right side: 5.937171044

Break this problem into two subproblems by setting 
(b + -3.5) equal to 5.937171044 and -5.937171044.

Subproblem 1

b + -3.5 = 5.937171044 Simplifying b + -3.5 = 5.937171044 Reorder the terms: -3.5 + b = 5.937171044 Solving -3.5 + b = 5.937171044 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '3.5' to each side of the equation. -3.5 + 3.5 + b = 5.937171044 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + b = 5.937171044 + 3.5 b = 5.937171044 + 3.5 Combine like terms: 5.937171044 + 3.5 = 9.437171044 b = 9.437171044 Simplifying b = 9.437171044

Subproblem 2

b + -3.5 = -5.937171044 Simplifying b + -3.5 = -5.937171044 Reorder the terms: -3.5 + b = -5.937171044 Solving -3.5 + b = -5.937171044 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '3.5' to each side of the equation. -3.5 + 3.5 + b = -5.937171044 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + b = -5.937171044 + 3.5 b = -5.937171044 + 3.5 Combine like terms: -5.937171044 + 3.5 = -2.437171044 b = -2.437171044 Simplifying b = -2.437171044

Solution

The solution to the problem is based on the solutions from the subproblems. b = {9.437171044, -2.437171044}

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