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Simplifying 7z + -6z3 + -3z3 + -1z2 = 0 Reorder the terms: 7z + -1z2 + -6z3 + -3z3 = 0 Combine like terms: -6z3 + -3z3 = -9z3 7z + -1z2 + -9z3 = 0 Solving 7z + -1z2 + -9z3 = 0 Solving for variable 'z'. Factor out the Greatest Common Factor (GCF), 'z'. z(7 + -1z + -9z2) = 0Subproblem 1
Set the factor 'z' equal to zero and attempt to solve: Simplifying z = 0 Solving z = 0 Move all terms containing z to the left, all other terms to the right. Simplifying z = 0Subproblem 2
Set the factor '(7 + -1z + -9z2)' equal to zero and attempt to solve: Simplifying 7 + -1z + -9z2 = 0 Solving 7 + -1z + -9z2 = 0 Begin completing the square. Divide all terms by -9 the coefficient of the squared term: Divide each side by '-9'. -0.7777777778 + 0.1111111111z + z2 = 0 Move the constant term to the right: Add '0.7777777778' to each side of the equation. -0.7777777778 + 0.1111111111z + 0.7777777778 + z2 = 0 + 0.7777777778 Reorder the terms: -0.7777777778 + 0.7777777778 + 0.1111111111z + z2 = 0 + 0.7777777778 Combine like terms: -0.7777777778 + 0.7777777778 = 0.0000000000 0.0000000000 + 0.1111111111z + z2 = 0 + 0.7777777778 0.1111111111z + z2 = 0 + 0.7777777778 Combine like terms: 0 + 0.7777777778 = 0.7777777778 0.1111111111z + z2 = 0.7777777778 The z term is 0.1111111111z. Take half its coefficient (0.05555555555). Square it (0.003086419752) and add it to both sides. Add '0.003086419752' to each side of the equation. 0.1111111111z + 0.003086419752 + z2 = 0.7777777778 + 0.003086419752 Reorder the terms: 0.003086419752 + 0.1111111111z + z2 = 0.7777777778 + 0.003086419752 Combine like terms: 0.7777777778 + 0.003086419752 = 0.780864197552 0.003086419752 + 0.1111111111z + z2 = 0.780864197552 Factor a perfect square on the left side: (z + 0.05555555555)(z + 0.05555555555) = 0.780864197552 Calculate the square root of the right side: 0.883665207 Break this problem into two subproblems by setting (z + 0.05555555555) equal to 0.883665207 and -0.883665207.Subproblem 1
z + 0.05555555555 = 0.883665207 Simplifying z + 0.05555555555 = 0.883665207 Reorder the terms: 0.05555555555 + z = 0.883665207 Solving 0.05555555555 + z = 0.883665207 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.05555555555' to each side of the equation. 0.05555555555 + -0.05555555555 + z = 0.883665207 + -0.05555555555 Combine like terms: 0.05555555555 + -0.05555555555 = 0.00000000000 0.00000000000 + z = 0.883665207 + -0.05555555555 z = 0.883665207 + -0.05555555555 Combine like terms: 0.883665207 + -0.05555555555 = 0.82810965145 z = 0.82810965145 Simplifying z = 0.82810965145Subproblem 2
z + 0.05555555555 = -0.883665207 Simplifying z + 0.05555555555 = -0.883665207 Reorder the terms: 0.05555555555 + z = -0.883665207 Solving 0.05555555555 + z = -0.883665207 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.05555555555' to each side of the equation. 0.05555555555 + -0.05555555555 + z = -0.883665207 + -0.05555555555 Combine like terms: 0.05555555555 + -0.05555555555 = 0.00000000000 0.00000000000 + z = -0.883665207 + -0.05555555555 z = -0.883665207 + -0.05555555555 Combine like terms: -0.883665207 + -0.05555555555 = -0.93922076255 z = -0.93922076255 Simplifying z = -0.93922076255Solution
The solution to the problem is based on the solutions from the subproblems. z = {0.82810965145, -0.93922076255}Solution
z = {0, 0.82810965145, -0.93922076255}
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