209.952=q(q-2)

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Solution for 209.952=q(q-2) equation:


Simplifying
209.952 = q(q + -2)

Reorder the terms:
209.952 = q(-2 + q)
209.952 = (-2 * q + q * q)
209.952 = (-2q + q2)

Solving
209.952 = -2q + q2

Solving for variable 'q'.

Reorder the terms:
209.952 + 2q + -1q2 = -2q + 2q + q2 + -1q2

Combine like terms: -2q + 2q = 0
209.952 + 2q + -1q2 = 0 + q2 + -1q2
209.952 + 2q + -1q2 = q2 + -1q2

Combine like terms: q2 + -1q2 = 0
209.952 + 2q + -1q2 = 0

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

Divide each side by '-1'.
-209.952 + -2q + q2 = 0

Move the constant term to the right:

Add '209.952' to each side of the equation.
-209.952 + -2q + 209.952 + q2 = 0 + 209.952

Reorder the terms:
-209.952 + 209.952 + -2q + q2 = 0 + 209.952

Combine like terms: -209.952 + 209.952 = 0.000
0.000 + -2q + q2 = 0 + 209.952
-2q + q2 = 0 + 209.952

Combine like terms: 0 + 209.952 = 209.952
-2q + q2 = 209.952

The q term is -2q.  Take half its coefficient (-1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
-2q + 1 + q2 = 209.952 + 1

Reorder the terms:
1 + -2q + q2 = 209.952 + 1

Combine like terms: 209.952 + 1 = 210.952
1 + -2q + q2 = 210.952

Factor a perfect square on the left side:
(q + -1)(q + -1) = 210.952

Calculate the square root of the right side: 14.524186724

Break this problem into two subproblems by setting 
(q + -1) equal to 14.524186724 and -14.524186724.

Subproblem 1

q + -1 = 14.524186724 Simplifying q + -1 = 14.524186724 Reorder the terms: -1 + q = 14.524186724 Solving -1 + q = 14.524186724 Solving for variable 'q'. Move all terms containing q to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + q = 14.524186724 + 1 Combine like terms: -1 + 1 = 0 0 + q = 14.524186724 + 1 q = 14.524186724 + 1 Combine like terms: 14.524186724 + 1 = 15.524186724 q = 15.524186724 Simplifying q = 15.524186724

Subproblem 2

q + -1 = -14.524186724 Simplifying q + -1 = -14.524186724 Reorder the terms: -1 + q = -14.524186724 Solving -1 + q = -14.524186724 Solving for variable 'q'. Move all terms containing q to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + q = -14.524186724 + 1 Combine like terms: -1 + 1 = 0 0 + q = -14.524186724 + 1 q = -14.524186724 + 1 Combine like terms: -14.524186724 + 1 = -13.524186724 q = -13.524186724 Simplifying q = -13.524186724

Solution

The solution to the problem is based on the solutions from the subproblems. q = {15.524186724, -13.524186724}

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