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Simplifying 3z2 + 12z + 5 = 0 Reorder the terms: 5 + 12z + 3z2 = 0 Solving 5 + 12z + 3z2 = 0 Solving for variable 'z'. Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. 1.666666667 + 4z + z2 = 0 Move the constant term to the right: Add '-1.666666667' to each side of the equation. 1.666666667 + 4z + -1.666666667 + z2 = 0 + -1.666666667 Reorder the terms: 1.666666667 + -1.666666667 + 4z + z2 = 0 + -1.666666667 Combine like terms: 1.666666667 + -1.666666667 = 0.000000000 0.000000000 + 4z + z2 = 0 + -1.666666667 4z + z2 = 0 + -1.666666667 Combine like terms: 0 + -1.666666667 = -1.666666667 4z + z2 = -1.666666667 The z term is 4z. Take half its coefficient (2). Square it (4) and add it to both sides. Add '4' to each side of the equation. 4z + 4 + z2 = -1.666666667 + 4 Reorder the terms: 4 + 4z + z2 = -1.666666667 + 4 Combine like terms: -1.666666667 + 4 = 2.333333333 4 + 4z + z2 = 2.333333333 Factor a perfect square on the left side: (z + 2)(z + 2) = 2.333333333 Calculate the square root of the right side: 1.527525232 Break this problem into two subproblems by setting (z + 2) equal to 1.527525232 and -1.527525232.Subproblem 1
z + 2 = 1.527525232 Simplifying z + 2 = 1.527525232 Reorder the terms: 2 + z = 1.527525232 Solving 2 + z = 1.527525232 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-2' to each side of the equation. 2 + -2 + z = 1.527525232 + -2 Combine like terms: 2 + -2 = 0 0 + z = 1.527525232 + -2 z = 1.527525232 + -2 Combine like terms: 1.527525232 + -2 = -0.472474768 z = -0.472474768 Simplifying z = -0.472474768Subproblem 2
z + 2 = -1.527525232 Simplifying z + 2 = -1.527525232 Reorder the terms: 2 + z = -1.527525232 Solving 2 + z = -1.527525232 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-2' to each side of the equation. 2 + -2 + z = -1.527525232 + -2 Combine like terms: 2 + -2 = 0 0 + z = -1.527525232 + -2 z = -1.527525232 + -2 Combine like terms: -1.527525232 + -2 = -3.527525232 z = -3.527525232 Simplifying z = -3.527525232Solution
The solution to the problem is based on the solutions from the subproblems. z = {-0.472474768, -3.527525232}
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