z(6z+30)=(z-15)

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Solution for z(6z+30)=(z-15) equation:


Simplifying
z(6z + 30) = (z + -15)

Reorder the terms:
z(30 + 6z) = (z + -15)
(30 * z + 6z * z) = (z + -15)
(30z + 6z2) = (z + -15)

Reorder the terms:
30z + 6z2 = (-15 + z)

Remove parenthesis around (-15 + z)
30z + 6z2 = -15 + z

Solving
30z + 6z2 = -15 + z

Solving for variable 'z'.

Reorder the terms:
15 + 30z + -1z + 6z2 = -15 + z + 15 + -1z

Combine like terms: 30z + -1z = 29z
15 + 29z + 6z2 = -15 + z + 15 + -1z

Reorder the terms:
15 + 29z + 6z2 = -15 + 15 + z + -1z

Combine like terms: -15 + 15 = 0
15 + 29z + 6z2 = 0 + z + -1z
15 + 29z + 6z2 = z + -1z

Combine like terms: z + -1z = 0
15 + 29z + 6z2 = 0

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

Divide each side by '6'.
2.5 + 4.833333333z + z2 = 0

Move the constant term to the right:

Add '-2.5' to each side of the equation.
2.5 + 4.833333333z + -2.5 + z2 = 0 + -2.5

Reorder the terms:
2.5 + -2.5 + 4.833333333z + z2 = 0 + -2.5

Combine like terms: 2.5 + -2.5 = 0.0
0.0 + 4.833333333z + z2 = 0 + -2.5
4.833333333z + z2 = 0 + -2.5

Combine like terms: 0 + -2.5 = -2.5
4.833333333z + z2 = -2.5

The z term is 4.833333333z.  Take half its coefficient (2.416666667).
Square it (5.840277779) and add it to both sides.

Add '5.840277779' to each side of the equation.
4.833333333z + 5.840277779 + z2 = -2.5 + 5.840277779

Reorder the terms:
5.840277779 + 4.833333333z + z2 = -2.5 + 5.840277779

Combine like terms: -2.5 + 5.840277779 = 3.340277779
5.840277779 + 4.833333333z + z2 = 3.340277779

Factor a perfect square on the left side:
(z + 2.416666667)(z + 2.416666667) = 3.340277779

Calculate the square root of the right side: 1.827642684

Break this problem into two subproblems by setting 
(z + 2.416666667) equal to 1.827642684 and -1.827642684.

Subproblem 1

z + 2.416666667 = 1.827642684 Simplifying z + 2.416666667 = 1.827642684 Reorder the terms: 2.416666667 + z = 1.827642684 Solving 2.416666667 + z = 1.827642684 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-2.416666667' to each side of the equation. 2.416666667 + -2.416666667 + z = 1.827642684 + -2.416666667 Combine like terms: 2.416666667 + -2.416666667 = 0.000000000 0.000000000 + z = 1.827642684 + -2.416666667 z = 1.827642684 + -2.416666667 Combine like terms: 1.827642684 + -2.416666667 = -0.589023983 z = -0.589023983 Simplifying z = -0.589023983

Subproblem 2

z + 2.416666667 = -1.827642684 Simplifying z + 2.416666667 = -1.827642684 Reorder the terms: 2.416666667 + z = -1.827642684 Solving 2.416666667 + z = -1.827642684 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-2.416666667' to each side of the equation. 2.416666667 + -2.416666667 + z = -1.827642684 + -2.416666667 Combine like terms: 2.416666667 + -2.416666667 = 0.000000000 0.000000000 + z = -1.827642684 + -2.416666667 z = -1.827642684 + -2.416666667 Combine like terms: -1.827642684 + -2.416666667 = -4.244309351 z = -4.244309351 Simplifying z = -4.244309351

Solution

The solution to the problem is based on the solutions from the subproblems. z = {-0.589023983, -4.244309351}

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