5z(4z+3)=150

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Solution for 5z(4z+3)=150 equation: 


Simplifying
5z(4z + 3) = 150

Reorder the terms:
5z(3 + 4z) = 150
(3 * 5z + 4z * 5z) = 150
(15z + 20z2) = 150

Solving
15z + 20z2 = 150

Solving for variable 'z'.

Reorder the terms:
-150 + 15z + 20z2 = 150 + -150

Combine like terms: 150 + -150 = 0
-150 + 15z + 20z2 = 0

Factor out the Greatest Common Factor (GCF), '5'.
5(-30 + 3z + 4z2) = 0

Ignore the factor 5.

Subproblem 1
Set the factor '(-30 + 3z + 4z2)' equal to zero and attempt to solve:

Simplifying
-30 + 3z + 4z2 = 0

Solving
-30 + 3z + 4z2 = 0

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

Divide each side by '4'.
-7.5 + 0.75z + z2 = 0

Move the constant term to the right:

Add '7.5' to each side of the equation.
-7.5 + 0.75z + 7.5 + z2 = 0 + 7.5

Reorder the terms:
-7.5 + 7.5 + 0.75z + z2 = 0 + 7.5

Combine like terms: -7.5 + 7.5 = 0.0
0.0 + 0.75z + z2 = 0 + 7.5
0.75z + z2 = 0 + 7.5

Combine like terms: 0 + 7.5 = 7.5
0.75z + z2 = 7.5

The z term is 0.75z.  Take half its coefficient (0.375).
Square it (0.140625) and add it to both sides.

Add '0.140625' to each side of the equation.
0.75z + 0.140625 + z2 = 7.5 + 0.140625

Reorder the terms:
0.140625 + 0.75z + z2 = 7.5 + 0.140625

Combine like terms: 7.5 + 0.140625 = 7.640625
0.140625 + 0.75z + z2 = 7.640625

Factor a perfect square on the left side:
(z + 0.375)(z + 0.375) = 7.640625

Calculate the square root of the right side: 2.764168048

Break this problem into two subproblems by setting 
(z + 0.375) equal to 2.764168048 and -2.764168048.


Subproblem 1
z + 0.375 = 2.764168048

Simplifying
z + 0.375 = 2.764168048

Reorder the terms:
0.375 + z = 2.764168048

Solving
0.375 + z = 2.764168048

Solving for variable 'z'.

Move all terms containing z to the left, all other terms to the right.

Add '-0.375' to each side of the equation.
0.375 + -0.375 + z = 2.764168048 + -0.375

Combine like terms: 0.375 + -0.375 = 0.000
0.000 + z = 2.764168048 + -0.375
z = 2.764168048 + -0.375

Combine like terms: 2.764168048 + -0.375 = 2.389168048
z = 2.389168048

Simplifying
z = 2.389168048


Subproblem 2
z + 0.375 = -2.764168048

Simplifying
z + 0.375 = -2.764168048

Reorder the terms:
0.375 + z = -2.764168048

Solving
0.375 + z = -2.764168048

Solving for variable 'z'.

Move all terms containing z to the left, all other terms to the right.

Add '-0.375' to each side of the equation.
0.375 + -0.375 + z = -2.764168048 + -0.375

Combine like terms: 0.375 + -0.375 = 0.000
0.000 + z = -2.764168048 + -0.375
z = -2.764168048 + -0.375

Combine like terms: -2.764168048 + -0.375 = -3.139168048
z = -3.139168048

Simplifying
z = -3.139168048


Solution
The solution to the problem is based on the solutions
from the subproblems.
z = {2.389168048, -3.139168048}
Solution
z = {2.389168048, -3.139168048}

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