-4q^2-84q=-16

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Solution for -4q^2-84q=-16 equation:


Simplifying
-4q2 + -84q = -16

Reorder the terms:
-84q + -4q2 = -16

Solving
-84q + -4q2 = -16

Solving for variable 'q'.

Reorder the terms:
16 + -84q + -4q2 = -16 + 16

Combine like terms: -16 + 16 = 0
16 + -84q + -4q2 = 0

Factor out the Greatest Common Factor (GCF), '4'.
4(4 + -21q + -1q2) = 0

Ignore the factor 4.

Subproblem 1
Set the factor '(4 + -21q + -1q2)' equal to zero and attempt to solve:

Simplifying
4 + -21q + -1q2 = 0

Solving
4 + -21q + -1q2 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
-4 + 21q + q2 = 0

Move the constant term to the right:

Add '4' to each side of the equation.
-4 + 21q + 4 + q2 = 0 + 4

Reorder the terms:
-4 + 4 + 21q + q2 = 0 + 4

Combine like terms: -4 + 4 = 0
0 + 21q + q2 = 0 + 4
21q + q2 = 0 + 4

Combine like terms: 0 + 4 = 4
21q + q2 = 4

The q term is 21q.  Take half its coefficient (10.5).
Square it (110.25) and add it to both sides.

Add '110.25' to each side of the equation.
21q + 110.25 + q2 = 4 + 110.25

Reorder the terms:
110.25 + 21q + q2 = 4 + 110.25

Combine like terms: 4 + 110.25 = 114.25
110.25 + 21q + q2 = 114.25

Factor a perfect square on the left side:
(q + 10.5)(q + 10.5) = 114.25

Calculate the square root of the right side: 10.688779163

Break this problem into two subproblems by setting 
(q + 10.5) equal to 10.688779163 and -10.688779163.


Subproblem 1
q + 10.5 = 10.688779163

Simplifying
q + 10.5 = 10.688779163

Reorder the terms:
10.5 + q = 10.688779163

Solving
10.5 + q = 10.688779163

Solving for variable 'q'.

Move all terms containing q to the left, all other terms to the right.

Add '-10.5' to each side of the equation.
10.5 + -10.5 + q = 10.688779163 + -10.5

Combine like terms: 10.5 + -10.5 = 0.0
0.0 + q = 10.688779163 + -10.5
q = 10.688779163 + -10.5

Combine like terms: 10.688779163 + -10.5 = 0.188779163
q = 0.188779163

Simplifying
q = 0.188779163


Subproblem 2
q + 10.5 = -10.688779163

Simplifying
q + 10.5 = -10.688779163

Reorder the terms:
10.5 + q = -10.688779163

Solving
10.5 + q = -10.688779163

Solving for variable 'q'.

Move all terms containing q to the left, all other terms to the right.

Add '-10.5' to each side of the equation.
10.5 + -10.5 + q = -10.688779163 + -10.5

Combine like terms: 10.5 + -10.5 = 0.0
0.0 + q = -10.688779163 + -10.5
q = -10.688779163 + -10.5

Combine like terms: -10.688779163 + -10.5 = -21.188779163
q = -21.188779163

Simplifying
q = -21.188779163


Solution
The solution to the problem is based on the solutions
from the subproblems.
q = {0.188779163, -21.188779163}
Solution
q = {0.188779163, -21.188779163}

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