b^2+3b-972=0

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Solution for b^2+3b-972=0 equation:


Simplifying
b2 + 3b + -972 = 0

Reorder the terms:
-972 + 3b + b2 = 0

Solving
-972 + 3b + b2 = 0

Solving for variable 'b'.

Begin completing the square.

Move the constant term to the right:

Add '972' to each side of the equation.
-972 + 3b + 972 + b2 = 0 + 972

Reorder the terms:
-972 + 972 + 3b + b2 = 0 + 972

Combine like terms: -972 + 972 = 0
0 + 3b + b2 = 0 + 972
3b + b2 = 0 + 972

Combine like terms: 0 + 972 = 972
3b + b2 = 972

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 = 972 + 2.25

Reorder the terms:
2.25 + 3b + b2 = 972 + 2.25

Combine like terms: 972 + 2.25 = 974.25
2.25 + 3b + b2 = 974.25

Factor a perfect square on the left side:
(b + 1.5)(b + 1.5) = 974.25

Calculate the square root of the right side: 31.21297807

Break this problem into two subproblems by setting 
(b + 1.5) equal to 31.21297807 and -31.21297807.

Subproblem 1

b + 1.5 = 31.21297807 Simplifying b + 1.5 = 31.21297807 Reorder the terms: 1.5 + b = 31.21297807 Solving 1.5 + b = 31.21297807 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 = 31.21297807 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + b = 31.21297807 + -1.5 b = 31.21297807 + -1.5 Combine like terms: 31.21297807 + -1.5 = 29.71297807 b = 29.71297807 Simplifying b = 29.71297807

Subproblem 2

b + 1.5 = -31.21297807 Simplifying b + 1.5 = -31.21297807 Reorder the terms: 1.5 + b = -31.21297807 Solving 1.5 + b = -31.21297807 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 = -31.21297807 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + b = -31.21297807 + -1.5 b = -31.21297807 + -1.5 Combine like terms: -31.21297807 + -1.5 = -32.71297807 b = -32.71297807 Simplifying b = -32.71297807

Solution

The solution to the problem is based on the solutions from the subproblems. b = {29.71297807, -32.71297807}

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