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Simplifying -2z2 + -52z + 46 = 0 Reorder the terms: 46 + -52z + -2z2 = 0 Solving 46 + -52z + -2z2 = 0 Solving for variable 'z'. Factor out the Greatest Common Factor (GCF), '2'. 2(23 + -26z + -1z2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(23 + -26z + -1z2)' equal to zero and attempt to solve: Simplifying 23 + -26z + -1z2 = 0 Solving 23 + -26z + -1z2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -23 + 26z + z2 = 0 Move the constant term to the right: Add '23' to each side of the equation. -23 + 26z + 23 + z2 = 0 + 23 Reorder the terms: -23 + 23 + 26z + z2 = 0 + 23 Combine like terms: -23 + 23 = 0 0 + 26z + z2 = 0 + 23 26z + z2 = 0 + 23 Combine like terms: 0 + 23 = 23 26z + z2 = 23 The z term is 26z. Take half its coefficient (13). Square it (169) and add it to both sides. Add '169' to each side of the equation. 26z + 169 + z2 = 23 + 169 Reorder the terms: 169 + 26z + z2 = 23 + 169 Combine like terms: 23 + 169 = 192 169 + 26z + z2 = 192 Factor a perfect square on the left side: (z + 13)(z + 13) = 192 Calculate the square root of the right side: 13.856406461 Break this problem into two subproblems by setting (z + 13) equal to 13.856406461 and -13.856406461.Subproblem 1
z + 13 = 13.856406461 Simplifying z + 13 = 13.856406461 Reorder the terms: 13 + z = 13.856406461 Solving 13 + z = 13.856406461 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-13' to each side of the equation. 13 + -13 + z = 13.856406461 + -13 Combine like terms: 13 + -13 = 0 0 + z = 13.856406461 + -13 z = 13.856406461 + -13 Combine like terms: 13.856406461 + -13 = 0.856406461 z = 0.856406461 Simplifying z = 0.856406461Subproblem 2
z + 13 = -13.856406461 Simplifying z + 13 = -13.856406461 Reorder the terms: 13 + z = -13.856406461 Solving 13 + z = -13.856406461 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-13' to each side of the equation. 13 + -13 + z = -13.856406461 + -13 Combine like terms: 13 + -13 = 0 0 + z = -13.856406461 + -13 z = -13.856406461 + -13 Combine like terms: -13.856406461 + -13 = -26.856406461 z = -26.856406461 Simplifying z = -26.856406461Solution
The solution to the problem is based on the solutions from the subproblems. z = {0.856406461, -26.856406461}Solution
z = {0.856406461, -26.856406461}
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