5q-15=q^2+4q*2q-27

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Solution for 5q-15=q^2+4q*2q-27 equation:


Simplifying
5q + -15 = q2 + 4q * 2q + -27

Reorder the terms:
-15 + 5q = q2 + 4q * 2q + -27

Reorder the terms for easier multiplication:
-15 + 5q = q2 + 4 * 2q * q + -27

Multiply 4 * 2
-15 + 5q = q2 + 8q * q + -27

Multiply q * q
-15 + 5q = q2 + 8q2 + -27

Reorder the terms:
-15 + 5q = -27 + q2 + 8q2

Combine like terms: q2 + 8q2 = 9q2
-15 + 5q = -27 + 9q2

Solving
-15 + 5q = -27 + 9q2

Solving for variable 'q'.

Reorder the terms:
-15 + 27 + 5q + -9q2 = -27 + 9q2 + 27 + -9q2

Combine like terms: -15 + 27 = 12
12 + 5q + -9q2 = -27 + 9q2 + 27 + -9q2

Reorder the terms:
12 + 5q + -9q2 = -27 + 27 + 9q2 + -9q2

Combine like terms: -27 + 27 = 0
12 + 5q + -9q2 = 0 + 9q2 + -9q2
12 + 5q + -9q2 = 9q2 + -9q2

Combine like terms: 9q2 + -9q2 = 0
12 + 5q + -9q2 = 0

Begin completing the square.  Divide all terms by
-9 the coefficient of the squared term: 

Divide each side by '-9'.
-1.333333333 + -0.5555555556q + q2 = 0

Move the constant term to the right:

Add '1.333333333' to each side of the equation.
-1.333333333 + -0.5555555556q + 1.333333333 + q2 = 0 + 1.333333333

Reorder the terms:
-1.333333333 + 1.333333333 + -0.5555555556q + q2 = 0 + 1.333333333

Combine like terms: -1.333333333 + 1.333333333 = 0.000000000
0.000000000 + -0.5555555556q + q2 = 0 + 1.333333333
-0.5555555556q + q2 = 0 + 1.333333333

Combine like terms: 0 + 1.333333333 = 1.333333333
-0.5555555556q + q2 = 1.333333333

The q term is -0.5555555556q.  Take half its coefficient (-0.2777777778).
Square it (0.07716049384) and add it to both sides.

Add '0.07716049384' to each side of the equation.
-0.5555555556q + 0.07716049384 + q2 = 1.333333333 + 0.07716049384

Reorder the terms:
0.07716049384 + -0.5555555556q + q2 = 1.333333333 + 0.07716049384

Combine like terms: 1.333333333 + 0.07716049384 = 1.41049382684
0.07716049384 + -0.5555555556q + q2 = 1.41049382684

Factor a perfect square on the left side:
(q + -0.2777777778)(q + -0.2777777778) = 1.41049382684

Calculate the square root of the right side: 1.187642129

Break this problem into two subproblems by setting 
(q + -0.2777777778) equal to 1.187642129 and -1.187642129.

Subproblem 1

q + -0.2777777778 = 1.187642129 Simplifying q + -0.2777777778 = 1.187642129 Reorder the terms: -0.2777777778 + q = 1.187642129 Solving -0.2777777778 + q = 1.187642129 Solving for variable 'q'. Move all terms containing q to the left, all other terms to the right. Add '0.2777777778' to each side of the equation. -0.2777777778 + 0.2777777778 + q = 1.187642129 + 0.2777777778 Combine like terms: -0.2777777778 + 0.2777777778 = 0.0000000000 0.0000000000 + q = 1.187642129 + 0.2777777778 q = 1.187642129 + 0.2777777778 Combine like terms: 1.187642129 + 0.2777777778 = 1.4654199068 q = 1.4654199068 Simplifying q = 1.4654199068

Subproblem 2

q + -0.2777777778 = -1.187642129 Simplifying q + -0.2777777778 = -1.187642129 Reorder the terms: -0.2777777778 + q = -1.187642129 Solving -0.2777777778 + q = -1.187642129 Solving for variable 'q'. Move all terms containing q to the left, all other terms to the right. Add '0.2777777778' to each side of the equation. -0.2777777778 + 0.2777777778 + q = -1.187642129 + 0.2777777778 Combine like terms: -0.2777777778 + 0.2777777778 = 0.0000000000 0.0000000000 + q = -1.187642129 + 0.2777777778 q = -1.187642129 + 0.2777777778 Combine like terms: -1.187642129 + 0.2777777778 = -0.9098643512 q = -0.9098643512 Simplifying q = -0.9098643512

Solution

The solution to the problem is based on the solutions from the subproblems. q = {1.4654199068, -0.9098643512}

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