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Simplifying 4z2 + 8z + -18 = 0 Reorder the terms: -18 + 8z + 4z2 = 0 Solving -18 + 8z + 4z2 = 0 Solving for variable 'z'. Factor out the Greatest Common Factor (GCF), '2'. 2(-9 + 4z + 2z2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-9 + 4z + 2z2)' equal to zero and attempt to solve: Simplifying -9 + 4z + 2z2 = 0 Solving -9 + 4z + 2z2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -4.5 + 2z + z2 = 0 Move the constant term to the right: Add '4.5' to each side of the equation. -4.5 + 2z + 4.5 + z2 = 0 + 4.5 Reorder the terms: -4.5 + 4.5 + 2z + z2 = 0 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + 2z + z2 = 0 + 4.5 2z + z2 = 0 + 4.5 Combine like terms: 0 + 4.5 = 4.5 2z + z2 = 4.5 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 = 4.5 + 1 Reorder the terms: 1 + 2z + z2 = 4.5 + 1 Combine like terms: 4.5 + 1 = 5.5 1 + 2z + z2 = 5.5 Factor a perfect square on the left side: (z + 1)(z + 1) = 5.5 Calculate the square root of the right side: 2.34520788 Break this problem into two subproblems by setting (z + 1) equal to 2.34520788 and -2.34520788.Subproblem 1
z + 1 = 2.34520788 Simplifying z + 1 = 2.34520788 Reorder the terms: 1 + z = 2.34520788 Solving 1 + z = 2.34520788 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 = 2.34520788 + -1 Combine like terms: 1 + -1 = 0 0 + z = 2.34520788 + -1 z = 2.34520788 + -1 Combine like terms: 2.34520788 + -1 = 1.34520788 z = 1.34520788 Simplifying z = 1.34520788Subproblem 2
z + 1 = -2.34520788 Simplifying z + 1 = -2.34520788 Reorder the terms: 1 + z = -2.34520788 Solving 1 + z = -2.34520788 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 = -2.34520788 + -1 Combine like terms: 1 + -1 = 0 0 + z = -2.34520788 + -1 z = -2.34520788 + -1 Combine like terms: -2.34520788 + -1 = -3.34520788 z = -3.34520788 Simplifying z = -3.34520788Solution
The solution to the problem is based on the solutions from the subproblems. z = {1.34520788, -3.34520788}Solution
z = {1.34520788, -3.34520788}
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