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