46z^2+96z+15=0

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Solution for 46z^2+96z+15=0 equation:


Simplifying
46z2 + 96z + 15 = 0

Reorder the terms:
15 + 96z + 46z2 = 0

Solving
15 + 96z + 46z2 = 0

Solving for variable 'z'.

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

Divide each side by '46'.
0.3260869565 + 2.086956522z + z2 = 0

Move the constant term to the right:

Add '-0.3260869565' to each side of the equation.
0.3260869565 + 2.086956522z + -0.3260869565 + z2 = 0 + -0.3260869565

Reorder the terms:
0.3260869565 + -0.3260869565 + 2.086956522z + z2 = 0 + -0.3260869565

Combine like terms: 0.3260869565 + -0.3260869565 = 0.0000000000
0.0000000000 + 2.086956522z + z2 = 0 + -0.3260869565
2.086956522z + z2 = 0 + -0.3260869565

Combine like terms: 0 + -0.3260869565 = -0.3260869565
2.086956522z + z2 = -0.3260869565

The z term is 2.086956522z.  Take half its coefficient (1.043478261).
Square it (1.088846881) and add it to both sides.

Add '1.088846881' to each side of the equation.
2.086956522z + 1.088846881 + z2 = -0.3260869565 + 1.088846881

Reorder the terms:
1.088846881 + 2.086956522z + z2 = -0.3260869565 + 1.088846881

Combine like terms: -0.3260869565 + 1.088846881 = 0.7627599245
1.088846881 + 2.086956522z + z2 = 0.7627599245

Factor a perfect square on the left side:
(z + 1.043478261)(z + 1.043478261) = 0.7627599245

Calculate the square root of the right side: 0.873361279

Break this problem into two subproblems by setting 
(z + 1.043478261) equal to 0.873361279 and -0.873361279.

Subproblem 1

z + 1.043478261 = 0.873361279 Simplifying z + 1.043478261 = 0.873361279 Reorder the terms: 1.043478261 + z = 0.873361279 Solving 1.043478261 + z = 0.873361279 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-1.043478261' to each side of the equation. 1.043478261 + -1.043478261 + z = 0.873361279 + -1.043478261 Combine like terms: 1.043478261 + -1.043478261 = 0.000000000 0.000000000 + z = 0.873361279 + -1.043478261 z = 0.873361279 + -1.043478261 Combine like terms: 0.873361279 + -1.043478261 = -0.170116982 z = -0.170116982 Simplifying z = -0.170116982

Subproblem 2

z + 1.043478261 = -0.873361279 Simplifying z + 1.043478261 = -0.873361279 Reorder the terms: 1.043478261 + z = -0.873361279 Solving 1.043478261 + z = -0.873361279 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-1.043478261' to each side of the equation. 1.043478261 + -1.043478261 + z = -0.873361279 + -1.043478261 Combine like terms: 1.043478261 + -1.043478261 = 0.000000000 0.000000000 + z = -0.873361279 + -1.043478261 z = -0.873361279 + -1.043478261 Combine like terms: -0.873361279 + -1.043478261 = -1.91683954 z = -1.91683954 Simplifying z = -1.91683954

Solution

The solution to the problem is based on the solutions from the subproblems. z = {-0.170116982, -1.91683954}

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