6q^2+3q=-2q+24

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Solution for 6q^2+3q=-2q+24 equation:


Simplifying
6q2 + 3q = -2q + 24

Reorder the terms:
3q + 6q2 = -2q + 24

Reorder the terms:
3q + 6q2 = 24 + -2q

Solving
3q + 6q2 = 24 + -2q

Solving for variable 'q'.

Reorder the terms:
-24 + 3q + 2q + 6q2 = 24 + -2q + -24 + 2q

Combine like terms: 3q + 2q = 5q
-24 + 5q + 6q2 = 24 + -2q + -24 + 2q

Reorder the terms:
-24 + 5q + 6q2 = 24 + -24 + -2q + 2q

Combine like terms: 24 + -24 = 0
-24 + 5q + 6q2 = 0 + -2q + 2q
-24 + 5q + 6q2 = -2q + 2q

Combine like terms: -2q + 2q = 0
-24 + 5q + 6q2 = 0

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

Divide each side by '6'.
-4 + 0.8333333333q + q2 = 0

Move the constant term to the right:

Add '4' to each side of the equation.
-4 + 0.8333333333q + 4 + q2 = 0 + 4

Reorder the terms:
-4 + 4 + 0.8333333333q + q2 = 0 + 4

Combine like terms: -4 + 4 = 0
0 + 0.8333333333q + q2 = 0 + 4
0.8333333333q + q2 = 0 + 4

Combine like terms: 0 + 4 = 4
0.8333333333q + q2 = 4

The q term is 0.8333333333q.  Take half its coefficient (0.4166666667).
Square it (0.1736111111) and add it to both sides.

Add '0.1736111111' to each side of the equation.
0.8333333333q + 0.1736111111 + q2 = 4 + 0.1736111111

Reorder the terms:
0.1736111111 + 0.8333333333q + q2 = 4 + 0.1736111111

Combine like terms: 4 + 0.1736111111 = 4.1736111111
0.1736111111 + 0.8333333333q + q2 = 4.1736111111

Factor a perfect square on the left side:
(q + 0.4166666667)(q + 0.4166666667) = 4.1736111111

Calculate the square root of the right side: 2.042941779

Break this problem into two subproblems by setting 
(q + 0.4166666667) equal to 2.042941779 and -2.042941779.

Subproblem 1

q + 0.4166666667 = 2.042941779 Simplifying q + 0.4166666667 = 2.042941779 Reorder the terms: 0.4166666667 + q = 2.042941779 Solving 0.4166666667 + q = 2.042941779 Solving for variable 'q'. Move all terms containing q to the left, all other terms to the right. Add '-0.4166666667' to each side of the equation. 0.4166666667 + -0.4166666667 + q = 2.042941779 + -0.4166666667 Combine like terms: 0.4166666667 + -0.4166666667 = 0.0000000000 0.0000000000 + q = 2.042941779 + -0.4166666667 q = 2.042941779 + -0.4166666667 Combine like terms: 2.042941779 + -0.4166666667 = 1.6262751123 q = 1.6262751123 Simplifying q = 1.6262751123

Subproblem 2

q + 0.4166666667 = -2.042941779 Simplifying q + 0.4166666667 = -2.042941779 Reorder the terms: 0.4166666667 + q = -2.042941779 Solving 0.4166666667 + q = -2.042941779 Solving for variable 'q'. Move all terms containing q to the left, all other terms to the right. Add '-0.4166666667' to each side of the equation. 0.4166666667 + -0.4166666667 + q = -2.042941779 + -0.4166666667 Combine like terms: 0.4166666667 + -0.4166666667 = 0.0000000000 0.0000000000 + q = -2.042941779 + -0.4166666667 q = -2.042941779 + -0.4166666667 Combine like terms: -2.042941779 + -0.4166666667 = -2.4596084457 q = -2.4596084457 Simplifying q = -2.4596084457

Solution

The solution to the problem is based on the solutions from the subproblems. q = {1.6262751123, -2.4596084457}

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