86-6q-3q^2=0.25q^2+10

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Solution for 86-6q-3q^2=0.25q^2+10 equation:


Simplifying
86 + -6q + -3q2 = 0.25q2 + 10

Reorder the terms:
86 + -6q + -3q2 = 10 + 0.25q2

Solving
86 + -6q + -3q2 = 10 + 0.25q2

Solving for variable 'q'.

Reorder the terms:
86 + -10 + -6q + -3q2 + -0.25q2 = 10 + 0.25q2 + -10 + -0.25q2

Combine like terms: 86 + -10 = 76
76 + -6q + -3q2 + -0.25q2 = 10 + 0.25q2 + -10 + -0.25q2

Combine like terms: -3q2 + -0.25q2 = -3.25q2
76 + -6q + -3.25q2 = 10 + 0.25q2 + -10 + -0.25q2

Reorder the terms:
76 + -6q + -3.25q2 = 10 + -10 + 0.25q2 + -0.25q2

Combine like terms: 10 + -10 = 0
76 + -6q + -3.25q2 = 0 + 0.25q2 + -0.25q2
76 + -6q + -3.25q2 = 0.25q2 + -0.25q2

Combine like terms: 0.25q2 + -0.25q2 = 0.00
76 + -6q + -3.25q2 = 0.00

Begin completing the square.  Divide all terms by
-3.25 the coefficient of the squared term: 

Divide each side by '-3.25'.
-23.38461538 + 1.846153846q + q2 = 0

Move the constant term to the right:

Add '23.38461538' to each side of the equation.
-23.38461538 + 1.846153846q + 23.38461538 + q2 = 0 + 23.38461538

Reorder the terms:
-23.38461538 + 23.38461538 + 1.846153846q + q2 = 0 + 23.38461538

Combine like terms: -23.38461538 + 23.38461538 = 0.00000000
0.00000000 + 1.846153846q + q2 = 0 + 23.38461538
1.846153846q + q2 = 0 + 23.38461538

Combine like terms: 0 + 23.38461538 = 23.38461538
1.846153846q + q2 = 23.38461538

The q term is 1.846153846q.  Take half its coefficient (0.923076923).
Square it (0.8520710058) and add it to both sides.

Add '0.8520710058' to each side of the equation.
1.846153846q + 0.8520710058 + q2 = 23.38461538 + 0.8520710058

Reorder the terms:
0.8520710058 + 1.846153846q + q2 = 23.38461538 + 0.8520710058

Combine like terms: 23.38461538 + 0.8520710058 = 24.2366863858
0.8520710058 + 1.846153846q + q2 = 24.2366863858

Factor a perfect square on the left side:
(q + 0.923076923)(q + 0.923076923) = 24.2366863858

Calculate the square root of the right side: 4.923076923

Break this problem into two subproblems by setting 
(q + 0.923076923) equal to 4.923076923 and -4.923076923.

Subproblem 1

q + 0.923076923 = 4.923076923 Simplifying q + 0.923076923 = 4.923076923 Reorder the terms: 0.923076923 + q = 4.923076923 Solving 0.923076923 + q = 4.923076923 Solving for variable 'q'. Move all terms containing q to the left, all other terms to the right. Add '-0.923076923' to each side of the equation. 0.923076923 + -0.923076923 + q = 4.923076923 + -0.923076923 Combine like terms: 0.923076923 + -0.923076923 = 0.000000000 0.000000000 + q = 4.923076923 + -0.923076923 q = 4.923076923 + -0.923076923 Combine like terms: 4.923076923 + -0.923076923 = 4 q = 4 Simplifying q = 4

Subproblem 2

q + 0.923076923 = -4.923076923 Simplifying q + 0.923076923 = -4.923076923 Reorder the terms: 0.923076923 + q = -4.923076923 Solving 0.923076923 + q = -4.923076923 Solving for variable 'q'. Move all terms containing q to the left, all other terms to the right. Add '-0.923076923' to each side of the equation. 0.923076923 + -0.923076923 + q = -4.923076923 + -0.923076923 Combine like terms: 0.923076923 + -0.923076923 = 0.000000000 0.000000000 + q = -4.923076923 + -0.923076923 q = -4.923076923 + -0.923076923 Combine like terms: -4.923076923 + -0.923076923 = -5.846153846 q = -5.846153846 Simplifying q = -5.846153846

Solution

The solution to the problem is based on the solutions from the subproblems. q = {4, -5.846153846}

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