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Simplifying 6z2 + 12z + -4 = 0 Reorder the terms: -4 + 12z + 6z2 = 0 Solving -4 + 12z + 6z2 = 0 Solving for variable 'z'. Factor out the Greatest Common Factor (GCF), '2'. 2(-2 + 6z + 3z2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-2 + 6z + 3z2)' equal to zero and attempt to solve: Simplifying -2 + 6z + 3z2 = 0 Solving -2 + 6z + 3z2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -0.6666666667 + 2z + z2 = 0 Move the constant term to the right: Add '0.6666666667' to each side of the equation. -0.6666666667 + 2z + 0.6666666667 + z2 = 0 + 0.6666666667 Reorder the terms: -0.6666666667 + 0.6666666667 + 2z + z2 = 0 + 0.6666666667 Combine like terms: -0.6666666667 + 0.6666666667 = 0.0000000000 0.0000000000 + 2z + z2 = 0 + 0.6666666667 2z + z2 = 0 + 0.6666666667 Combine like terms: 0 + 0.6666666667 = 0.6666666667 2z + z2 = 0.6666666667 The z term is 2z. Take half its coefficient (1). Square it (1) and add it to both sides. Add '1' to each side of the equation. 2z + 1 + z2 = 0.6666666667 + 1 Reorder the terms: 1 + 2z + z2 = 0.6666666667 + 1 Combine like terms: 0.6666666667 + 1 = 1.6666666667 1 + 2z + z2 = 1.6666666667 Factor a perfect square on the left side: (z + 1)(z + 1) = 1.6666666667 Calculate the square root of the right side: 1.290994449 Break this problem into two subproblems by setting (z + 1) equal to 1.290994449 and -1.290994449.Subproblem 1
z + 1 = 1.290994449 Simplifying z + 1 = 1.290994449 Reorder the terms: 1 + z = 1.290994449 Solving 1 + z = 1.290994449 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + z = 1.290994449 + -1 Combine like terms: 1 + -1 = 0 0 + z = 1.290994449 + -1 z = 1.290994449 + -1 Combine like terms: 1.290994449 + -1 = 0.290994449 z = 0.290994449 Simplifying z = 0.290994449Subproblem 2
z + 1 = -1.290994449 Simplifying z + 1 = -1.290994449 Reorder the terms: 1 + z = -1.290994449 Solving 1 + z = -1.290994449 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + z = -1.290994449 + -1 Combine like terms: 1 + -1 = 0 0 + z = -1.290994449 + -1 z = -1.290994449 + -1 Combine like terms: -1.290994449 + -1 = -2.290994449 z = -2.290994449 Simplifying z = -2.290994449Solution
The solution to the problem is based on the solutions from the subproblems. z = {0.290994449, -2.290994449}Solution
z = {0.290994449, -2.290994449}
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