-16z^2+24z+60=0

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Solution for -16z^2+24z+60=0 equation: 


Simplifying
-16z2 + 24z + 60 = 0

Reorder the terms:
60 + 24z + -16z2 = 0

Solving
60 + 24z + -16z2 = 0

Solving for variable 'z'.

Factor out the Greatest Common Factor (GCF), '4'.
4(15 + 6z + -4z2) = 0

Ignore the factor 4.

Subproblem 1
Set the factor '(15 + 6z + -4z2)' equal to zero and attempt to solve:

Simplifying
15 + 6z + -4z2 = 0

Solving
15 + 6z + -4z2 = 0

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

Divide each side by '-4'.
-3.75 + -1.5z + z2 = 0

Move the constant term to the right:

Add '3.75' to each side of the equation.
-3.75 + -1.5z + 3.75 + z2 = 0 + 3.75

Reorder the terms:
-3.75 + 3.75 + -1.5z + z2 = 0 + 3.75

Combine like terms: -3.75 + 3.75 = 0.00
0.00 + -1.5z + z2 = 0 + 3.75
-1.5z + z2 = 0 + 3.75

Combine like terms: 0 + 3.75 = 3.75
-1.5z + z2 = 3.75

The z term is -1.5z.  Take half its coefficient (-0.75).
Square it (0.5625) and add it to both sides.

Add '0.5625' to each side of the equation.
-1.5z + 0.5625 + z2 = 3.75 + 0.5625

Reorder the terms:
0.5625 + -1.5z + z2 = 3.75 + 0.5625

Combine like terms: 3.75 + 0.5625 = 4.3125
0.5625 + -1.5z + z2 = 4.3125

Factor a perfect square on the left side:
(z + -0.75)(z + -0.75) = 4.3125

Calculate the square root of the right side: 2.076655966

Break this problem into two subproblems by setting 
(z + -0.75) equal to 2.076655966 and -2.076655966.


Subproblem 1
z + -0.75 = 2.076655966

Simplifying
z + -0.75 = 2.076655966

Reorder the terms:
-0.75 + z = 2.076655966

Solving
-0.75 + z = 2.076655966

Solving for variable 'z'.

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

Add '0.75' to each side of the equation.
-0.75 + 0.75 + z = 2.076655966 + 0.75

Combine like terms: -0.75 + 0.75 = 0.00
0.00 + z = 2.076655966 + 0.75
z = 2.076655966 + 0.75

Combine like terms: 2.076655966 + 0.75 = 2.826655966
z = 2.826655966

Simplifying
z = 2.826655966


Subproblem 2
z + -0.75 = -2.076655966

Simplifying
z + -0.75 = -2.076655966

Reorder the terms:
-0.75 + z = -2.076655966

Solving
-0.75 + z = -2.076655966

Solving for variable 'z'.

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

Add '0.75' to each side of the equation.
-0.75 + 0.75 + z = -2.076655966 + 0.75

Combine like terms: -0.75 + 0.75 = 0.00
0.00 + z = -2.076655966 + 0.75
z = -2.076655966 + 0.75

Combine like terms: -2.076655966 + 0.75 = -1.326655966
z = -1.326655966

Simplifying
z = -1.326655966


Solution
The solution to the problem is based on the solutions
from the subproblems.
z = {2.826655966, -1.326655966}
Solution
z = {2.826655966, -1.326655966}

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