15z^2+7z-16=0

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Solution for 15z^2+7z-16=0 equation: 


Simplifying
15z2 + 7z + -16 = 0

Reorder the terms:
-16 + 7z + 15z2 = 0

Solving
-16 + 7z + 15z2 = 0

Solving for variable 'z'.

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

Divide each side by '15'.
-1.066666667 + 0.4666666667z + z2 = 0

Move the constant term to the right:

Add '1.066666667' to each side of the equation.
-1.066666667 + 0.4666666667z + 1.066666667 + z2 = 0 + 1.066666667

Reorder the terms:
-1.066666667 + 1.066666667 + 0.4666666667z + z2 = 0 + 1.066666667

Combine like terms: -1.066666667 + 1.066666667 = 0.000000000
0.000000000 + 0.4666666667z + z2 = 0 + 1.066666667
0.4666666667z + z2 = 0 + 1.066666667

Combine like terms: 0 + 1.066666667 = 1.066666667
0.4666666667z + z2 = 1.066666667

The z term is 0.4666666667z.  Take half its coefficient (0.2333333334).
Square it (0.05444444448) and add it to both sides.

Add '0.05444444448' to each side of the equation.
0.4666666667z + 0.05444444448 + z2 = 1.066666667 + 0.05444444448

Reorder the terms:
0.05444444448 + 0.4666666667z + z2 = 1.066666667 + 0.05444444448

Combine like terms: 1.066666667 + 0.05444444448 = 1.12111111148
0.05444444448 + 0.4666666667z + z2 = 1.12111111148

Factor a perfect square on the left side:
(z + 0.2333333334)(z + 0.2333333334) = 1.12111111148

Calculate the square root of the right side: 1.058825345

Break this problem into two subproblems by setting 
(z + 0.2333333334) equal to 1.058825345 and -1.058825345.


Subproblem 1
z + 0.2333333334 = 1.058825345

Simplifying
z + 0.2333333334 = 1.058825345

Reorder the terms:
0.2333333334 + z = 1.058825345

Solving
0.2333333334 + z = 1.058825345

Solving for variable 'z'.

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

Add '-0.2333333334' to each side of the equation.
0.2333333334 + -0.2333333334 + z = 1.058825345 + -0.2333333334

Combine like terms: 0.2333333334 + -0.2333333334 = 0.0000000000
0.0000000000 + z = 1.058825345 + -0.2333333334
z = 1.058825345 + -0.2333333334

Combine like terms: 1.058825345 + -0.2333333334 = 0.8254920116
z = 0.8254920116

Simplifying
z = 0.8254920116


Subproblem 2
z + 0.2333333334 = -1.058825345

Simplifying
z + 0.2333333334 = -1.058825345

Reorder the terms:
0.2333333334 + z = -1.058825345

Solving
0.2333333334 + z = -1.058825345

Solving for variable 'z'.

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

Add '-0.2333333334' to each side of the equation.
0.2333333334 + -0.2333333334 + z = -1.058825345 + -0.2333333334

Combine like terms: 0.2333333334 + -0.2333333334 = 0.0000000000
0.0000000000 + z = -1.058825345 + -0.2333333334
z = -1.058825345 + -0.2333333334

Combine like terms: -1.058825345 + -0.2333333334 = -1.2921586784
z = -1.2921586784

Simplifying
z = -1.2921586784


Solution
The solution to the problem is based on the solutions
from the subproblems.
z = {0.8254920116, -1.2921586784}

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