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Simplifying 18z2 + -8z = 2 Reorder the terms: -8z + 18z2 = 2 Solving -8z + 18z2 = 2 Solving for variable 'z'. Reorder the terms: -2 + -8z + 18z2 = 2 + -2 Combine like terms: 2 + -2 = 0 -2 + -8z + 18z2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-1 + -4z + 9z2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-1 + -4z + 9z2)' equal to zero and attempt to solve: Simplifying -1 + -4z + 9z2 = 0 Solving -1 + -4z + 9z2 = 0 Begin completing the square. Divide all terms by 9 the coefficient of the squared term: Divide each side by '9'. -0.1111111111 + -0.4444444444z + z2 = 0 Move the constant term to the right: Add '0.1111111111' to each side of the equation. -0.1111111111 + -0.4444444444z + 0.1111111111 + z2 = 0 + 0.1111111111 Reorder the terms: -0.1111111111 + 0.1111111111 + -0.4444444444z + z2 = 0 + 0.1111111111 Combine like terms: -0.1111111111 + 0.1111111111 = 0.0000000000 0.0000000000 + -0.4444444444z + z2 = 0 + 0.1111111111 -0.4444444444z + z2 = 0 + 0.1111111111 Combine like terms: 0 + 0.1111111111 = 0.1111111111 -0.4444444444z + z2 = 0.1111111111 The z term is -0.4444444444z. Take half its coefficient (-0.2222222222). Square it (0.04938271604) and add it to both sides. Add '0.04938271604' to each side of the equation. -0.4444444444z + 0.04938271604 + z2 = 0.1111111111 + 0.04938271604 Reorder the terms: 0.04938271604 + -0.4444444444z + z2 = 0.1111111111 + 0.04938271604 Combine like terms: 0.1111111111 + 0.04938271604 = 0.16049382714 0.04938271604 + -0.4444444444z + z2 = 0.16049382714 Factor a perfect square on the left side: (z + -0.2222222222)(z + -0.2222222222) = 0.16049382714 Calculate the square root of the right side: 0.400616808 Break this problem into two subproblems by setting (z + -0.2222222222) equal to 0.400616808 and -0.400616808.Subproblem 1
z + -0.2222222222 = 0.400616808 Simplifying z + -0.2222222222 = 0.400616808 Reorder the terms: -0.2222222222 + z = 0.400616808 Solving -0.2222222222 + z = 0.400616808 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.2222222222' to each side of the equation. -0.2222222222 + 0.2222222222 + z = 0.400616808 + 0.2222222222 Combine like terms: -0.2222222222 + 0.2222222222 = 0.0000000000 0.0000000000 + z = 0.400616808 + 0.2222222222 z = 0.400616808 + 0.2222222222 Combine like terms: 0.400616808 + 0.2222222222 = 0.6228390302 z = 0.6228390302 Simplifying z = 0.6228390302Subproblem 2
z + -0.2222222222 = -0.400616808 Simplifying z + -0.2222222222 = -0.400616808 Reorder the terms: -0.2222222222 + z = -0.400616808 Solving -0.2222222222 + z = -0.400616808 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.2222222222' to each side of the equation. -0.2222222222 + 0.2222222222 + z = -0.400616808 + 0.2222222222 Combine like terms: -0.2222222222 + 0.2222222222 = 0.0000000000 0.0000000000 + z = -0.400616808 + 0.2222222222 z = -0.400616808 + 0.2222222222 Combine like terms: -0.400616808 + 0.2222222222 = -0.1783945858 z = -0.1783945858 Simplifying z = -0.1783945858Solution
The solution to the problem is based on the solutions from the subproblems. z = {0.6228390302, -0.1783945858}Solution
z = {0.6228390302, -0.1783945858}
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