-10z+3(5+4z)=z(3z+3)

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Solution for -10z+3(5+4z)=z(3z+3) equation:


Simplifying
-10z + 3(5 + 4z) = z(3z + 3)
-10z + (5 * 3 + 4z * 3) = z(3z + 3)
-10z + (15 + 12z) = z(3z + 3)

Reorder the terms:
15 + -10z + 12z = z(3z + 3)

Combine like terms: -10z + 12z = 2z
15 + 2z = z(3z + 3)

Reorder the terms:
15 + 2z = z(3 + 3z)
15 + 2z = (3 * z + 3z * z)
15 + 2z = (3z + 3z2)

Solving
15 + 2z = 3z + 3z2

Solving for variable 'z'.

Combine like terms: 2z + -3z = -1z
15 + -1z + -3z2 = 3z + 3z2 + -3z + -3z2

Reorder the terms:
15 + -1z + -3z2 = 3z + -3z + 3z2 + -3z2

Combine like terms: 3z + -3z = 0
15 + -1z + -3z2 = 0 + 3z2 + -3z2
15 + -1z + -3z2 = 3z2 + -3z2

Combine like terms: 3z2 + -3z2 = 0
15 + -1z + -3z2 = 0

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

Divide each side by '-3'.
-5 + 0.3333333333z + z2 = 0

Move the constant term to the right:

Add '5' to each side of the equation.
-5 + 0.3333333333z + 5 + z2 = 0 + 5

Reorder the terms:
-5 + 5 + 0.3333333333z + z2 = 0 + 5

Combine like terms: -5 + 5 = 0
0 + 0.3333333333z + z2 = 0 + 5
0.3333333333z + z2 = 0 + 5

Combine like terms: 0 + 5 = 5
0.3333333333z + z2 = 5

The z term is 0.3333333333z.  Take half its coefficient (0.1666666667).
Square it (0.02777777779) and add it to both sides.

Add '0.02777777779' to each side of the equation.
0.3333333333z + 0.02777777779 + z2 = 5 + 0.02777777779

Reorder the terms:
0.02777777779 + 0.3333333333z + z2 = 5 + 0.02777777779

Combine like terms: 5 + 0.02777777779 = 5.02777777779
0.02777777779 + 0.3333333333z + z2 = 5.02777777779

Factor a perfect square on the left side:
(z + 0.1666666667)(z + 0.1666666667) = 5.02777777779

Calculate the square root of the right side: 2.242270675

Break this problem into two subproblems by setting 
(z + 0.1666666667) equal to 2.242270675 and -2.242270675.

Subproblem 1

z + 0.1666666667 = 2.242270675 Simplifying z + 0.1666666667 = 2.242270675 Reorder the terms: 0.1666666667 + z = 2.242270675 Solving 0.1666666667 + z = 2.242270675 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.1666666667' to each side of the equation. 0.1666666667 + -0.1666666667 + z = 2.242270675 + -0.1666666667 Combine like terms: 0.1666666667 + -0.1666666667 = 0.0000000000 0.0000000000 + z = 2.242270675 + -0.1666666667 z = 2.242270675 + -0.1666666667 Combine like terms: 2.242270675 + -0.1666666667 = 2.0756040083 z = 2.0756040083 Simplifying z = 2.0756040083

Subproblem 2

z + 0.1666666667 = -2.242270675 Simplifying z + 0.1666666667 = -2.242270675 Reorder the terms: 0.1666666667 + z = -2.242270675 Solving 0.1666666667 + z = -2.242270675 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.1666666667' to each side of the equation. 0.1666666667 + -0.1666666667 + z = -2.242270675 + -0.1666666667 Combine like terms: 0.1666666667 + -0.1666666667 = 0.0000000000 0.0000000000 + z = -2.242270675 + -0.1666666667 z = -2.242270675 + -0.1666666667 Combine like terms: -2.242270675 + -0.1666666667 = -2.4089373417 z = -2.4089373417 Simplifying z = -2.4089373417

Solution

The solution to the problem is based on the solutions from the subproblems. z = {2.0756040083, -2.4089373417}

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