16z^2-24z-90=0

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Solution for 16z^2-24z-90=0 equation: 


Simplifying
16z2 + -24z + -90 = 0

Reorder the terms:
-90 + -24z + 16z2 = 0

Solving
-90 + -24z + 16z2 = 0

Solving for variable 'z'.

Factor out the Greatest Common Factor (GCF), '2'.
2(-45 + -12z + 8z2) = 0

Ignore the factor 2.

Subproblem 1
Set the factor '(-45 + -12z + 8z2)' equal to zero and attempt to solve:

Simplifying
-45 + -12z + 8z2 = 0

Solving
-45 + -12z + 8z2 = 0

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

Divide each side by '8'.
-5.625 + -1.5z + z2 = 0

Move the constant term to the right:

Add '5.625' to each side of the equation.
-5.625 + -1.5z + 5.625 + z2 = 0 + 5.625

Reorder the terms:
-5.625 + 5.625 + -1.5z + z2 = 0 + 5.625

Combine like terms: -5.625 + 5.625 = 0.000
0.000 + -1.5z + z2 = 0 + 5.625
-1.5z + z2 = 0 + 5.625

Combine like terms: 0 + 5.625 = 5.625
-1.5z + z2 = 5.625

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 = 5.625 + 0.5625

Reorder the terms:
0.5625 + -1.5z + z2 = 5.625 + 0.5625

Combine like terms: 5.625 + 0.5625 = 6.1875
0.5625 + -1.5z + z2 = 6.1875

Factor a perfect square on the left side:
(z + -0.75)(z + -0.75) = 6.1875

Calculate the square root of the right side: 2.487468593

Break this problem into two subproblems by setting 
(z + -0.75) equal to 2.487468593 and -2.487468593.


Subproblem 1
z + -0.75 = 2.487468593

Simplifying
z + -0.75 = 2.487468593

Reorder the terms:
-0.75 + z = 2.487468593

Solving
-0.75 + z = 2.487468593

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 = 2.487468593 + 0.75

Combine like terms: -0.75 + 0.75 = 0.00
0.00 + z = 2.487468593 + 0.75
z = 2.487468593 + 0.75

Combine like terms: 2.487468593 + 0.75 = 3.237468593
z = 3.237468593

Simplifying
z = 3.237468593


Subproblem 2
z + -0.75 = -2.487468593

Simplifying
z + -0.75 = -2.487468593

Reorder the terms:
-0.75 + z = -2.487468593

Solving
-0.75 + z = -2.487468593

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 = -2.487468593 + 0.75

Combine like terms: -0.75 + 0.75 = 0.00
0.00 + z = -2.487468593 + 0.75
z = -2.487468593 + 0.75

Combine like terms: -2.487468593 + 0.75 = -1.737468593
z = -1.737468593

Simplifying
z = -1.737468593


Solution
The solution to the problem is based on the solutions
from the subproblems.
z = {3.237468593, -1.737468593}
Solution
z = {3.237468593, -1.737468593}

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