38z^2+96z+15=0

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Solution for 38z^2+96z+15=0 equation:


Simplifying
38z2 + 96z + 15 = 0

Reorder the terms:
15 + 96z + 38z2 = 0

Solving
15 + 96z + 38z2 = 0

Solving for variable 'z'.

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

Divide each side by '38'.
0.3947368421 + 2.526315789z + z2 = 0

Move the constant term to the right:

Add '-0.3947368421' to each side of the equation.
0.3947368421 + 2.526315789z + -0.3947368421 + z2 = 0 + -0.3947368421

Reorder the terms:
0.3947368421 + -0.3947368421 + 2.526315789z + z2 = 0 + -0.3947368421

Combine like terms: 0.3947368421 + -0.3947368421 = 0.0000000000
0.0000000000 + 2.526315789z + z2 = 0 + -0.3947368421
2.526315789z + z2 = 0 + -0.3947368421

Combine like terms: 0 + -0.3947368421 = -0.3947368421
2.526315789z + z2 = -0.3947368421

The z term is 2.526315789z.  Take half its coefficient (1.263157895).
Square it (1.595567868) and add it to both sides.

Add '1.595567868' to each side of the equation.
2.526315789z + 1.595567868 + z2 = -0.3947368421 + 1.595567868

Reorder the terms:
1.595567868 + 2.526315789z + z2 = -0.3947368421 + 1.595567868

Combine like terms: -0.3947368421 + 1.595567868 = 1.2008310259
1.595567868 + 2.526315789z + z2 = 1.2008310259

Factor a perfect square on the left side:
(z + 1.263157895)(z + 1.263157895) = 1.2008310259

Calculate the square root of the right side: 1.095824359

Break this problem into two subproblems by setting 
(z + 1.263157895) equal to 1.095824359 and -1.095824359.

Subproblem 1

z + 1.263157895 = 1.095824359 Simplifying z + 1.263157895 = 1.095824359 Reorder the terms: 1.263157895 + z = 1.095824359 Solving 1.263157895 + z = 1.095824359 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-1.263157895' to each side of the equation. 1.263157895 + -1.263157895 + z = 1.095824359 + -1.263157895 Combine like terms: 1.263157895 + -1.263157895 = 0.000000000 0.000000000 + z = 1.095824359 + -1.263157895 z = 1.095824359 + -1.263157895 Combine like terms: 1.095824359 + -1.263157895 = -0.167333536 z = -0.167333536 Simplifying z = -0.167333536

Subproblem 2

z + 1.263157895 = -1.095824359 Simplifying z + 1.263157895 = -1.095824359 Reorder the terms: 1.263157895 + z = -1.095824359 Solving 1.263157895 + z = -1.095824359 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-1.263157895' to each side of the equation. 1.263157895 + -1.263157895 + z = -1.095824359 + -1.263157895 Combine like terms: 1.263157895 + -1.263157895 = 0.000000000 0.000000000 + z = -1.095824359 + -1.263157895 z = -1.095824359 + -1.263157895 Combine like terms: -1.095824359 + -1.263157895 = -2.358982254 z = -2.358982254 Simplifying z = -2.358982254

Solution

The solution to the problem is based on the solutions from the subproblems. z = {-0.167333536, -2.358982254}

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