34-1.6666q-0.3333q^2-2q^2+3q+71=0

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Solution for 34-1.6666q-0.3333q^2-2q^2+3q+71=0 equation:


Simplifying
34 + -1.6666q + -0.3333q2 + -2q2 + 3q + 71 = 0

Reorder the terms:
34 + 71 + -1.6666q + 3q + -0.3333q2 + -2q2 = 0

Combine like terms: 34 + 71 = 105
105 + -1.6666q + 3q + -0.3333q2 + -2q2 = 0

Combine like terms: -1.6666q + 3q = 1.3334q
105 + 1.3334q + -0.3333q2 + -2q2 = 0

Combine like terms: -0.3333q2 + -2q2 = -2.3333q2
105 + 1.3334q + -2.3333q2 = 0

Solving
105 + 1.3334q + -2.3333q2 = 0

Solving for variable 'q'.

Begin completing the square.  Divide all terms by
-2.3333 the coefficient of the squared term: 

Divide each side by '-2.3333'.
-45.00064287 + -0.5714653066q + q2 = 0

Move the constant term to the right:

Add '45.00064287' to each side of the equation.
-45.00064287 + -0.5714653066q + 45.00064287 + q2 = 0 + 45.00064287

Reorder the terms:
-45.00064287 + 45.00064287 + -0.5714653066q + q2 = 0 + 45.00064287

Combine like terms: -45.00064287 + 45.00064287 = 0.00000000
0.00000000 + -0.5714653066q + q2 = 0 + 45.00064287
-0.5714653066q + q2 = 0 + 45.00064287

Combine like terms: 0 + 45.00064287 = 45.00064287
-0.5714653066q + q2 = 45.00064287

The q term is -0.5714653066q.  Take half its coefficient (-0.2857326533).
Square it (0.08164314916) and add it to both sides.

Add '0.08164314916' to each side of the equation.
-0.5714653066q + 0.08164314916 + q2 = 45.00064287 + 0.08164314916

Reorder the terms:
0.08164314916 + -0.5714653066q + q2 = 45.00064287 + 0.08164314916

Combine like terms: 45.00064287 + 0.08164314916 = 45.08228601916
0.08164314916 + -0.5714653066q + q2 = 45.08228601916

Factor a perfect square on the left side:
(q + -0.2857326533)(q + -0.2857326533) = 45.08228601916

Calculate the square root of the right side: 6.714334369

Break this problem into two subproblems by setting 
(q + -0.2857326533) equal to 6.714334369 and -6.714334369.

Subproblem 1

q + -0.2857326533 = 6.714334369 Simplifying q + -0.2857326533 = 6.714334369 Reorder the terms: -0.2857326533 + q = 6.714334369 Solving -0.2857326533 + q = 6.714334369 Solving for variable 'q'. Move all terms containing q to the left, all other terms to the right. Add '0.2857326533' to each side of the equation. -0.2857326533 + 0.2857326533 + q = 6.714334369 + 0.2857326533 Combine like terms: -0.2857326533 + 0.2857326533 = 0.0000000000 0.0000000000 + q = 6.714334369 + 0.2857326533 q = 6.714334369 + 0.2857326533 Combine like terms: 6.714334369 + 0.2857326533 = 7.0000670223 q = 7.0000670223 Simplifying q = 7.0000670223

Subproblem 2

q + -0.2857326533 = -6.714334369 Simplifying q + -0.2857326533 = -6.714334369 Reorder the terms: -0.2857326533 + q = -6.714334369 Solving -0.2857326533 + q = -6.714334369 Solving for variable 'q'. Move all terms containing q to the left, all other terms to the right. Add '0.2857326533' to each side of the equation. -0.2857326533 + 0.2857326533 + q = -6.714334369 + 0.2857326533 Combine like terms: -0.2857326533 + 0.2857326533 = 0.0000000000 0.0000000000 + q = -6.714334369 + 0.2857326533 q = -6.714334369 + 0.2857326533 Combine like terms: -6.714334369 + 0.2857326533 = -6.4286017157 q = -6.4286017157 Simplifying q = -6.4286017157

Solution

The solution to the problem is based on the solutions from the subproblems. q = {7.0000670223, -6.4286017157}

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