9q+3q(q-7)=18-q

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Solution for 9q+3q(q-7)=18-q equation:


Simplifying
9q + 3q(q + -7) = 18 + -1q

Reorder the terms:
9q + 3q(-7 + q) = 18 + -1q
9q + (-7 * 3q + q * 3q) = 18 + -1q
9q + (-21q + 3q2) = 18 + -1q

Combine like terms: 9q + -21q = -12q
-12q + 3q2 = 18 + -1q

Solving
-12q + 3q2 = 18 + -1q

Solving for variable 'q'.

Reorder the terms:
-18 + -12q + q + 3q2 = 18 + -1q + -18 + q

Combine like terms: -12q + q = -11q
-18 + -11q + 3q2 = 18 + -1q + -18 + q

Reorder the terms:
-18 + -11q + 3q2 = 18 + -18 + -1q + q

Combine like terms: 18 + -18 = 0
-18 + -11q + 3q2 = 0 + -1q + q
-18 + -11q + 3q2 = -1q + q

Combine like terms: -1q + q = 0
-18 + -11q + 3q2 = 0

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

Divide each side by '3'.
-6 + -3.666666667q + q2 = 0

Move the constant term to the right:

Add '6' to each side of the equation.
-6 + -3.666666667q + 6 + q2 = 0 + 6

Reorder the terms:
-6 + 6 + -3.666666667q + q2 = 0 + 6

Combine like terms: -6 + 6 = 0
0 + -3.666666667q + q2 = 0 + 6
-3.666666667q + q2 = 0 + 6

Combine like terms: 0 + 6 = 6
-3.666666667q + q2 = 6

The q term is -3.666666667q.  Take half its coefficient (-1.833333334).
Square it (3.361111114) and add it to both sides.

Add '3.361111114' to each side of the equation.
-3.666666667q + 3.361111114 + q2 = 6 + 3.361111114

Reorder the terms:
3.361111114 + -3.666666667q + q2 = 6 + 3.361111114

Combine like terms: 6 + 3.361111114 = 9.361111114
3.361111114 + -3.666666667q + q2 = 9.361111114

Factor a perfect square on the left side:
(q + -1.833333334)(q + -1.833333334) = 9.361111114

Calculate the square root of the right side: 3.059593292

Break this problem into two subproblems by setting 
(q + -1.833333334) equal to 3.059593292 and -3.059593292.

Subproblem 1

q + -1.833333334 = 3.059593292 Simplifying q + -1.833333334 = 3.059593292 Reorder the terms: -1.833333334 + q = 3.059593292 Solving -1.833333334 + q = 3.059593292 Solving for variable 'q'. Move all terms containing q to the left, all other terms to the right. Add '1.833333334' to each side of the equation. -1.833333334 + 1.833333334 + q = 3.059593292 + 1.833333334 Combine like terms: -1.833333334 + 1.833333334 = 0.000000000 0.000000000 + q = 3.059593292 + 1.833333334 q = 3.059593292 + 1.833333334 Combine like terms: 3.059593292 + 1.833333334 = 4.892926626 q = 4.892926626 Simplifying q = 4.892926626

Subproblem 2

q + -1.833333334 = -3.059593292 Simplifying q + -1.833333334 = -3.059593292 Reorder the terms: -1.833333334 + q = -3.059593292 Solving -1.833333334 + q = -3.059593292 Solving for variable 'q'. Move all terms containing q to the left, all other terms to the right. Add '1.833333334' to each side of the equation. -1.833333334 + 1.833333334 + q = -3.059593292 + 1.833333334 Combine like terms: -1.833333334 + 1.833333334 = 0.000000000 0.000000000 + q = -3.059593292 + 1.833333334 q = -3.059593292 + 1.833333334 Combine like terms: -3.059593292 + 1.833333334 = -1.226259958 q = -1.226259958 Simplifying q = -1.226259958

Solution

The solution to the problem is based on the solutions from the subproblems. q = {4.892926626, -1.226259958}

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