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Simplifying 2z2 + -19z + 3 = 0 Reorder the terms: 3 + -19z + 2z2 = 0 Solving 3 + -19z + 2z2 = 0 Solving for variable 'z'. Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. 1.5 + -9.5z + z2 = 0 Move the constant term to the right: Add '-1.5' to each side of the equation. 1.5 + -9.5z + -1.5 + z2 = 0 + -1.5 Reorder the terms: 1.5 + -1.5 + -9.5z + z2 = 0 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + -9.5z + z2 = 0 + -1.5 -9.5z + z2 = 0 + -1.5 Combine like terms: 0 + -1.5 = -1.5 -9.5z + z2 = -1.5 The z term is -9.5z. Take half its coefficient (-4.75). Square it (22.5625) and add it to both sides. Add '22.5625' to each side of the equation. -9.5z + 22.5625 + z2 = -1.5 + 22.5625 Reorder the terms: 22.5625 + -9.5z + z2 = -1.5 + 22.5625 Combine like terms: -1.5 + 22.5625 = 21.0625 22.5625 + -9.5z + z2 = 21.0625 Factor a perfect square on the left side: (z + -4.75)(z + -4.75) = 21.0625 Calculate the square root of the right side: 4.589389938 Break this problem into two subproblems by setting (z + -4.75) equal to 4.589389938 and -4.589389938.Subproblem 1
z + -4.75 = 4.589389938 Simplifying z + -4.75 = 4.589389938 Reorder the terms: -4.75 + z = 4.589389938 Solving -4.75 + z = 4.589389938 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '4.75' to each side of the equation. -4.75 + 4.75 + z = 4.589389938 + 4.75 Combine like terms: -4.75 + 4.75 = 0.00 0.00 + z = 4.589389938 + 4.75 z = 4.589389938 + 4.75 Combine like terms: 4.589389938 + 4.75 = 9.339389938 z = 9.339389938 Simplifying z = 9.339389938Subproblem 2
z + -4.75 = -4.589389938 Simplifying z + -4.75 = -4.589389938 Reorder the terms: -4.75 + z = -4.589389938 Solving -4.75 + z = -4.589389938 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '4.75' to each side of the equation. -4.75 + 4.75 + z = -4.589389938 + 4.75 Combine like terms: -4.75 + 4.75 = 0.00 0.00 + z = -4.589389938 + 4.75 z = -4.589389938 + 4.75 Combine like terms: -4.589389938 + 4.75 = 0.160610062 z = 0.160610062 Simplifying z = 0.160610062Solution
The solution to the problem is based on the solutions from the subproblems. z = {9.339389938, 0.160610062}
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