3q(q-5)=2(q+5)

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Solution for 3q(q-5)=2(q+5) equation:


Simplifying
3q(q + -5) = 2(q + 5)

Reorder the terms:
3q(-5 + q) = 2(q + 5)
(-5 * 3q + q * 3q) = 2(q + 5)
(-15q + 3q2) = 2(q + 5)

Reorder the terms:
-15q + 3q2 = 2(5 + q)
-15q + 3q2 = (5 * 2 + q * 2)
-15q + 3q2 = (10 + 2q)

Solving
-15q + 3q2 = 10 + 2q

Solving for variable 'q'.

Reorder the terms:
-10 + -15q + -2q + 3q2 = 10 + 2q + -10 + -2q

Combine like terms: -15q + -2q = -17q
-10 + -17q + 3q2 = 10 + 2q + -10 + -2q

Reorder the terms:
-10 + -17q + 3q2 = 10 + -10 + 2q + -2q

Combine like terms: 10 + -10 = 0
-10 + -17q + 3q2 = 0 + 2q + -2q
-10 + -17q + 3q2 = 2q + -2q

Combine like terms: 2q + -2q = 0
-10 + -17q + 3q2 = 0

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

Divide each side by '3'.
-3.333333333 + -5.666666667q + q2 = 0

Move the constant term to the right:

Add '3.333333333' to each side of the equation.
-3.333333333 + -5.666666667q + 3.333333333 + q2 = 0 + 3.333333333

Reorder the terms:
-3.333333333 + 3.333333333 + -5.666666667q + q2 = 0 + 3.333333333

Combine like terms: -3.333333333 + 3.333333333 = 0.000000000
0.000000000 + -5.666666667q + q2 = 0 + 3.333333333
-5.666666667q + q2 = 0 + 3.333333333

Combine like terms: 0 + 3.333333333 = 3.333333333
-5.666666667q + q2 = 3.333333333

The q term is -5.666666667q.  Take half its coefficient (-2.833333334).
Square it (8.027777782) and add it to both sides.

Add '8.027777782' to each side of the equation.
-5.666666667q + 8.027777782 + q2 = 3.333333333 + 8.027777782

Reorder the terms:
8.027777782 + -5.666666667q + q2 = 3.333333333 + 8.027777782

Combine like terms: 3.333333333 + 8.027777782 = 11.361111115
8.027777782 + -5.666666667q + q2 = 11.361111115

Factor a perfect square on the left side:
(q + -2.833333334)(q + -2.833333334) = 11.361111115

Calculate the square root of the right side: 3.370624737

Break this problem into two subproblems by setting 
(q + -2.833333334) equal to 3.370624737 and -3.370624737.

Subproblem 1

q + -2.833333334 = 3.370624737 Simplifying q + -2.833333334 = 3.370624737 Reorder the terms: -2.833333334 + q = 3.370624737 Solving -2.833333334 + q = 3.370624737 Solving for variable 'q'. Move all terms containing q to the left, all other terms to the right. Add '2.833333334' to each side of the equation. -2.833333334 + 2.833333334 + q = 3.370624737 + 2.833333334 Combine like terms: -2.833333334 + 2.833333334 = 0.000000000 0.000000000 + q = 3.370624737 + 2.833333334 q = 3.370624737 + 2.833333334 Combine like terms: 3.370624737 + 2.833333334 = 6.203958071 q = 6.203958071 Simplifying q = 6.203958071

Subproblem 2

q + -2.833333334 = -3.370624737 Simplifying q + -2.833333334 = -3.370624737 Reorder the terms: -2.833333334 + q = -3.370624737 Solving -2.833333334 + q = -3.370624737 Solving for variable 'q'. Move all terms containing q to the left, all other terms to the right. Add '2.833333334' to each side of the equation. -2.833333334 + 2.833333334 + q = -3.370624737 + 2.833333334 Combine like terms: -2.833333334 + 2.833333334 = 0.000000000 0.000000000 + q = -3.370624737 + 2.833333334 q = -3.370624737 + 2.833333334 Combine like terms: -3.370624737 + 2.833333334 = -0.537291403 q = -0.537291403 Simplifying q = -0.537291403

Solution

The solution to the problem is based on the solutions from the subproblems. q = {6.203958071, -0.537291403}

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