15z^2+z+2=0

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


Simplifying
15z2 + z + 2 = 0

Reorder the terms:
2 + z + 15z2 = 0

Solving
2 + z + 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'.
0.1333333333 + 0.06666666667z + z2 = 0

Move the constant term to the right:

Add '-0.1333333333' to each side of the equation.
0.1333333333 + 0.06666666667z + -0.1333333333 + z2 = 0 + -0.1333333333

Reorder the terms:
0.1333333333 + -0.1333333333 + 0.06666666667z + z2 = 0 + -0.1333333333

Combine like terms: 0.1333333333 + -0.1333333333 = 0.0000000000
0.0000000000 + 0.06666666667z + z2 = 0 + -0.1333333333
0.06666666667z + z2 = 0 + -0.1333333333

Combine like terms: 0 + -0.1333333333 = -0.1333333333
0.06666666667z + z2 = -0.1333333333

The z term is z.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
0.06666666667z + 0.25 + z2 = -0.1333333333 + 0.25

Reorder the terms:
0.25 + 0.06666666667z + z2 = -0.1333333333 + 0.25

Combine like terms: -0.1333333333 + 0.25 = 0.1166666667
0.25 + 0.06666666667z + z2 = 0.1166666667

Factor a perfect square on the left side:
(z + 0.5)(z + 0.5) = 0.1166666667

Calculate the square root of the right side: 0.341565026

Break this problem into two subproblems by setting 
(z + 0.5) equal to 0.341565026 and -0.341565026.

Subproblem 1

z + 0.5 = 0.341565026 Simplifying z + 0.5 = 0.341565026 Reorder the terms: 0.5 + z = 0.341565026 Solving 0.5 + z = 0.341565026 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + z = 0.341565026 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + z = 0.341565026 + -0.5 z = 0.341565026 + -0.5 Combine like terms: 0.341565026 + -0.5 = -0.158434974 z = -0.158434974 Simplifying z = -0.158434974

Subproblem 2

z + 0.5 = -0.341565026 Simplifying z + 0.5 = -0.341565026 Reorder the terms: 0.5 + z = -0.341565026 Solving 0.5 + z = -0.341565026 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + z = -0.341565026 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + z = -0.341565026 + -0.5 z = -0.341565026 + -0.5 Combine like terms: -0.341565026 + -0.5 = -0.841565026 z = -0.841565026 Simplifying z = -0.841565026

Solution

The solution to the problem is based on the solutions from the subproblems. z = {-0.158434974, -0.841565026}

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