36z^2+96z+5=0

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Solution for 36z^2+96z+5=0 equation:


Simplifying
36z2 + 96z + 5 = 0

Reorder the terms:
5 + 96z + 36z2 = 0

Solving
5 + 96z + 36z2 = 0

Solving for variable 'z'.

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

Divide each side by '36'.
0.1388888889 + 2.666666667z + z2 = 0

Move the constant term to the right:

Add '-0.1388888889' to each side of the equation.
0.1388888889 + 2.666666667z + -0.1388888889 + z2 = 0 + -0.1388888889

Reorder the terms:
0.1388888889 + -0.1388888889 + 2.666666667z + z2 = 0 + -0.1388888889

Combine like terms: 0.1388888889 + -0.1388888889 = 0.0000000000
0.0000000000 + 2.666666667z + z2 = 0 + -0.1388888889
2.666666667z + z2 = 0 + -0.1388888889

Combine like terms: 0 + -0.1388888889 = -0.1388888889
2.666666667z + z2 = -0.1388888889

The z term is 2.666666667z.  Take half its coefficient (1.333333334).
Square it (1.777777780) and add it to both sides.

Add '1.777777780' to each side of the equation.
2.666666667z + 1.777777780 + z2 = -0.1388888889 + 1.777777780

Reorder the terms:
1.777777780 + 2.666666667z + z2 = -0.1388888889 + 1.777777780

Combine like terms: -0.1388888889 + 1.777777780 = 1.6388888911
1.777777780 + 2.666666667z + z2 = 1.6388888911

Factor a perfect square on the left side:
(z + 1.333333334)(z + 1.333333334) = 1.6388888911

Calculate the square root of the right side: 1.280190959

Break this problem into two subproblems by setting 
(z + 1.333333334) equal to 1.280190959 and -1.280190959.

Subproblem 1

z + 1.333333334 = 1.280190959 Simplifying z + 1.333333334 = 1.280190959 Reorder the terms: 1.333333334 + z = 1.280190959 Solving 1.333333334 + z = 1.280190959 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-1.333333334' to each side of the equation. 1.333333334 + -1.333333334 + z = 1.280190959 + -1.333333334 Combine like terms: 1.333333334 + -1.333333334 = 0.000000000 0.000000000 + z = 1.280190959 + -1.333333334 z = 1.280190959 + -1.333333334 Combine like terms: 1.280190959 + -1.333333334 = -0.053142375 z = -0.053142375 Simplifying z = -0.053142375

Subproblem 2

z + 1.333333334 = -1.280190959 Simplifying z + 1.333333334 = -1.280190959 Reorder the terms: 1.333333334 + z = -1.280190959 Solving 1.333333334 + z = -1.280190959 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-1.333333334' to each side of the equation. 1.333333334 + -1.333333334 + z = -1.280190959 + -1.333333334 Combine like terms: 1.333333334 + -1.333333334 = 0.000000000 0.000000000 + z = -1.280190959 + -1.333333334 z = -1.280190959 + -1.333333334 Combine like terms: -1.280190959 + -1.333333334 = -2.613524293 z = -2.613524293 Simplifying z = -2.613524293

Solution

The solution to the problem is based on the solutions from the subproblems. z = {-0.053142375, -2.613524293}

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