q^2-27q=-96

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Solution for q^2-27q=-96 equation:


Simplifying
q2 + -27q = -96

Reorder the terms:
-27q + q2 = -96

Solving
-27q + q2 = -96

Solving for variable 'q'.

Reorder the terms:
96 + -27q + q2 = -96 + 96

Combine like terms: -96 + 96 = 0
96 + -27q + q2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-96' to each side of the equation.
96 + -27q + -96 + q2 = 0 + -96

Reorder the terms:
96 + -96 + -27q + q2 = 0 + -96

Combine like terms: 96 + -96 = 0
0 + -27q + q2 = 0 + -96
-27q + q2 = 0 + -96

Combine like terms: 0 + -96 = -96
-27q + q2 = -96

The q term is -27q.  Take half its coefficient (-13.5).
Square it (182.25) and add it to both sides.

Add '182.25' to each side of the equation.
-27q + 182.25 + q2 = -96 + 182.25

Reorder the terms:
182.25 + -27q + q2 = -96 + 182.25

Combine like terms: -96 + 182.25 = 86.25
182.25 + -27q + q2 = 86.25

Factor a perfect square on the left side:
(q + -13.5)(q + -13.5) = 86.25

Calculate the square root of the right side: 9.287087811

Break this problem into two subproblems by setting 
(q + -13.5) equal to 9.287087811 and -9.287087811.

Subproblem 1

q + -13.5 = 9.287087811 Simplifying q + -13.5 = 9.287087811 Reorder the terms: -13.5 + q = 9.287087811 Solving -13.5 + q = 9.287087811 Solving for variable 'q'. Move all terms containing q to the left, all other terms to the right. Add '13.5' to each side of the equation. -13.5 + 13.5 + q = 9.287087811 + 13.5 Combine like terms: -13.5 + 13.5 = 0.0 0.0 + q = 9.287087811 + 13.5 q = 9.287087811 + 13.5 Combine like terms: 9.287087811 + 13.5 = 22.787087811 q = 22.787087811 Simplifying q = 22.787087811

Subproblem 2

q + -13.5 = -9.287087811 Simplifying q + -13.5 = -9.287087811 Reorder the terms: -13.5 + q = -9.287087811 Solving -13.5 + q = -9.287087811 Solving for variable 'q'. Move all terms containing q to the left, all other terms to the right. Add '13.5' to each side of the equation. -13.5 + 13.5 + q = -9.287087811 + 13.5 Combine like terms: -13.5 + 13.5 = 0.0 0.0 + q = -9.287087811 + 13.5 q = -9.287087811 + 13.5 Combine like terms: -9.287087811 + 13.5 = 4.212912189 q = 4.212912189 Simplifying q = 4.212912189

Solution

The solution to the problem is based on the solutions from the subproblems. q = {22.787087811, 4.212912189}

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