0=4q^2+12q-9

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Solution for 0=4q^2+12q-9 equation:


Simplifying
0 = 4q2 + 12q + -9

Reorder the terms:
0 = -9 + 12q + 4q2

Solving
0 = -9 + 12q + 4q2

Solving for variable 'q'.

Combine like terms: 0 + 9 = 9
9 + -12q + -4q2 = -9 + 12q + 4q2 + 9 + -12q + -4q2

Reorder the terms:
9 + -12q + -4q2 = -9 + 9 + 12q + -12q + 4q2 + -4q2

Combine like terms: -9 + 9 = 0
9 + -12q + -4q2 = 0 + 12q + -12q + 4q2 + -4q2
9 + -12q + -4q2 = 12q + -12q + 4q2 + -4q2

Combine like terms: 12q + -12q = 0
9 + -12q + -4q2 = 0 + 4q2 + -4q2
9 + -12q + -4q2 = 4q2 + -4q2

Combine like terms: 4q2 + -4q2 = 0
9 + -12q + -4q2 = 0

Begin completing the square.  Divide all terms by
-4 the coefficient of the squared term: 

Divide each side by '-4'.
-2.25 + 3q + q2 = 0

Move the constant term to the right:

Add '2.25' to each side of the equation.
-2.25 + 3q + 2.25 + q2 = 0 + 2.25

Reorder the terms:
-2.25 + 2.25 + 3q + q2 = 0 + 2.25

Combine like terms: -2.25 + 2.25 = 0.00
0.00 + 3q + q2 = 0 + 2.25
3q + q2 = 0 + 2.25

Combine like terms: 0 + 2.25 = 2.25
3q + q2 = 2.25

The q term is 3q.  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.
3q + 2.25 + q2 = 2.25 + 2.25

Reorder the terms:
2.25 + 3q + q2 = 2.25 + 2.25

Combine like terms: 2.25 + 2.25 = 4.5
2.25 + 3q + q2 = 4.5

Factor a perfect square on the left side:
(q + 1.5)(q + 1.5) = 4.5

Calculate the square root of the right side: 2.121320344

Break this problem into two subproblems by setting 
(q + 1.5) equal to 2.121320344 and -2.121320344.

Subproblem 1

q + 1.5 = 2.121320344 Simplifying q + 1.5 = 2.121320344 Reorder the terms: 1.5 + q = 2.121320344 Solving 1.5 + q = 2.121320344 Solving for variable 'q'. Move all terms containing q to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + q = 2.121320344 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + q = 2.121320344 + -1.5 q = 2.121320344 + -1.5 Combine like terms: 2.121320344 + -1.5 = 0.621320344 q = 0.621320344 Simplifying q = 0.621320344

Subproblem 2

q + 1.5 = -2.121320344 Simplifying q + 1.5 = -2.121320344 Reorder the terms: 1.5 + q = -2.121320344 Solving 1.5 + q = -2.121320344 Solving for variable 'q'. Move all terms containing q to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + q = -2.121320344 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + q = -2.121320344 + -1.5 q = -2.121320344 + -1.5 Combine like terms: -2.121320344 + -1.5 = -3.621320344 q = -3.621320344 Simplifying q = -3.621320344

Solution

The solution to the problem is based on the solutions from the subproblems. q = {0.621320344, -3.621320344}

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