2q^2+12q-27=0

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Solution for 2q^2+12q-27=0 equation:


Simplifying
2q2 + 12q + -27 = 0

Reorder the terms:
-27 + 12q + 2q2 = 0

Solving
-27 + 12q + 2q2 = 0

Solving for variable 'q'.

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

Divide each side by '2'.
-13.5 + 6q + q2 = 0

Move the constant term to the right:

Add '13.5' to each side of the equation.
-13.5 + 6q + 13.5 + q2 = 0 + 13.5

Reorder the terms:
-13.5 + 13.5 + 6q + q2 = 0 + 13.5

Combine like terms: -13.5 + 13.5 = 0.0
0.0 + 6q + q2 = 0 + 13.5
6q + q2 = 0 + 13.5

Combine like terms: 0 + 13.5 = 13.5
6q + q2 = 13.5

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

Add '9' to each side of the equation.
6q + 9 + q2 = 13.5 + 9

Reorder the terms:
9 + 6q + q2 = 13.5 + 9

Combine like terms: 13.5 + 9 = 22.5
9 + 6q + q2 = 22.5

Factor a perfect square on the left side:
(q + 3)(q + 3) = 22.5

Calculate the square root of the right side: 4.74341649

Break this problem into two subproblems by setting 
(q + 3) equal to 4.74341649 and -4.74341649.

Subproblem 1

q + 3 = 4.74341649 Simplifying q + 3 = 4.74341649 Reorder the terms: 3 + q = 4.74341649 Solving 3 + q = 4.74341649 Solving for variable 'q'. Move all terms containing q to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + q = 4.74341649 + -3 Combine like terms: 3 + -3 = 0 0 + q = 4.74341649 + -3 q = 4.74341649 + -3 Combine like terms: 4.74341649 + -3 = 1.74341649 q = 1.74341649 Simplifying q = 1.74341649

Subproblem 2

q + 3 = -4.74341649 Simplifying q + 3 = -4.74341649 Reorder the terms: 3 + q = -4.74341649 Solving 3 + q = -4.74341649 Solving for variable 'q'. Move all terms containing q to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + q = -4.74341649 + -3 Combine like terms: 3 + -3 = 0 0 + q = -4.74341649 + -3 q = -4.74341649 + -3 Combine like terms: -4.74341649 + -3 = -7.74341649 q = -7.74341649 Simplifying q = -7.74341649

Solution

The solution to the problem is based on the solutions from the subproblems. q = {1.74341649, -7.74341649}

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