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Simplifying (-12z2 + -1z + 3) + (2z2 + 6z + -1) = 0 Reorder the terms: (3 + -1z + -12z2) + (2z2 + 6z + -1) = 0 Remove parenthesis around (3 + -1z + -12z2) 3 + -1z + -12z2 + (2z2 + 6z + -1) = 0 Reorder the terms: 3 + -1z + -12z2 + (-1 + 6z + 2z2) = 0 Remove parenthesis around (-1 + 6z + 2z2) 3 + -1z + -12z2 + -1 + 6z + 2z2 = 0 Reorder the terms: 3 + -1 + -1z + 6z + -12z2 + 2z2 = 0 Combine like terms: 3 + -1 = 2 2 + -1z + 6z + -12z2 + 2z2 = 0 Combine like terms: -1z + 6z = 5z 2 + 5z + -12z2 + 2z2 = 0 Combine like terms: -12z2 + 2z2 = -10z2 2 + 5z + -10z2 = 0 Solving 2 + 5z + -10z2 = 0 Solving for variable 'z'. Begin completing the square. Divide all terms by -10 the coefficient of the squared term: Divide each side by '-10'. -0.2 + -0.5z + z2 = 0.0 Move the constant term to the right: Add '0.2' to each side of the equation. -0.2 + -0.5z + 0.2 + z2 = 0.0 + 0.2 Reorder the terms: -0.2 + 0.2 + -0.5z + z2 = 0.0 + 0.2 Combine like terms: -0.2 + 0.2 = 0.0 0.0 + -0.5z + z2 = 0.0 + 0.2 -0.5z + z2 = 0.0 + 0.2 Combine like terms: 0.0 + 0.2 = 0.2 -0.5z + z2 = 0.2 The z term is -0.5z. Take half its coefficient (-0.25). Square it (0.0625) and add it to both sides. Add '0.0625' to each side of the equation. -0.5z + 0.0625 + z2 = 0.2 + 0.0625 Reorder the terms: 0.0625 + -0.5z + z2 = 0.2 + 0.0625 Combine like terms: 0.2 + 0.0625 = 0.2625 0.0625 + -0.5z + z2 = 0.2625 Factor a perfect square on the left side: (z + -0.25)(z + -0.25) = 0.2625 Calculate the square root of the right side: 0.512347538 Break this problem into two subproblems by setting (z + -0.25) equal to 0.512347538 and -0.512347538.Subproblem 1
z + -0.25 = 0.512347538 Simplifying z + -0.25 = 0.512347538 Reorder the terms: -0.25 + z = 0.512347538 Solving -0.25 + z = 0.512347538 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.25' to each side of the equation. -0.25 + 0.25 + z = 0.512347538 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + z = 0.512347538 + 0.25 z = 0.512347538 + 0.25 Combine like terms: 0.512347538 + 0.25 = 0.762347538 z = 0.762347538 Simplifying z = 0.762347538Subproblem 2
z + -0.25 = -0.512347538 Simplifying z + -0.25 = -0.512347538 Reorder the terms: -0.25 + z = -0.512347538 Solving -0.25 + z = -0.512347538 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.25' to each side of the equation. -0.25 + 0.25 + z = -0.512347538 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + z = -0.512347538 + 0.25 z = -0.512347538 + 0.25 Combine like terms: -0.512347538 + 0.25 = -0.262347538 z = -0.262347538 Simplifying z = -0.262347538Solution
The solution to the problem is based on the solutions from the subproblems. z = {0.762347538, -0.262347538}
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