85-3q=4q^2+6

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Solution for 85-3q=4q^2+6 equation:


Simplifying
85 + -3q = 4q2 + 6

Reorder the terms:
85 + -3q = 6 + 4q2

Solving
85 + -3q = 6 + 4q2

Solving for variable 'q'.

Reorder the terms:
85 + -6 + -3q + -4q2 = 6 + 4q2 + -6 + -4q2

Combine like terms: 85 + -6 = 79
79 + -3q + -4q2 = 6 + 4q2 + -6 + -4q2

Reorder the terms:
79 + -3q + -4q2 = 6 + -6 + 4q2 + -4q2

Combine like terms: 6 + -6 = 0
79 + -3q + -4q2 = 0 + 4q2 + -4q2
79 + -3q + -4q2 = 4q2 + -4q2

Combine like terms: 4q2 + -4q2 = 0
79 + -3q + -4q2 = 0

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

Divide each side by '-4'.
-19.75 + 0.75q + q2 = 0

Move the constant term to the right:

Add '19.75' to each side of the equation.
-19.75 + 0.75q + 19.75 + q2 = 0 + 19.75

Reorder the terms:
-19.75 + 19.75 + 0.75q + q2 = 0 + 19.75

Combine like terms: -19.75 + 19.75 = 0.00
0.00 + 0.75q + q2 = 0 + 19.75
0.75q + q2 = 0 + 19.75

Combine like terms: 0 + 19.75 = 19.75
0.75q + q2 = 19.75

The q term is 0.75q.  Take half its coefficient (0.375).
Square it (0.140625) and add it to both sides.

Add '0.140625' to each side of the equation.
0.75q + 0.140625 + q2 = 19.75 + 0.140625

Reorder the terms:
0.140625 + 0.75q + q2 = 19.75 + 0.140625

Combine like terms: 19.75 + 0.140625 = 19.890625
0.140625 + 0.75q + q2 = 19.890625

Factor a perfect square on the left side:
(q + 0.375)(q + 0.375) = 19.890625

Calculate the square root of the right side: 4.459890694

Break this problem into two subproblems by setting 
(q + 0.375) equal to 4.459890694 and -4.459890694.

Subproblem 1

q + 0.375 = 4.459890694 Simplifying q + 0.375 = 4.459890694 Reorder the terms: 0.375 + q = 4.459890694 Solving 0.375 + q = 4.459890694 Solving for variable 'q'. Move all terms containing q to the left, all other terms to the right. Add '-0.375' to each side of the equation. 0.375 + -0.375 + q = 4.459890694 + -0.375 Combine like terms: 0.375 + -0.375 = 0.000 0.000 + q = 4.459890694 + -0.375 q = 4.459890694 + -0.375 Combine like terms: 4.459890694 + -0.375 = 4.084890694 q = 4.084890694 Simplifying q = 4.084890694

Subproblem 2

q + 0.375 = -4.459890694 Simplifying q + 0.375 = -4.459890694 Reorder the terms: 0.375 + q = -4.459890694 Solving 0.375 + q = -4.459890694 Solving for variable 'q'. Move all terms containing q to the left, all other terms to the right. Add '-0.375' to each side of the equation. 0.375 + -0.375 + q = -4.459890694 + -0.375 Combine like terms: 0.375 + -0.375 = 0.000 0.000 + q = -4.459890694 + -0.375 q = -4.459890694 + -0.375 Combine like terms: -4.459890694 + -0.375 = -4.834890694 q = -4.834890694 Simplifying q = -4.834890694

Solution

The solution to the problem is based on the solutions from the subproblems. q = {4.084890694, -4.834890694}

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