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Simplifying 2z + 5 = 4z2 Reorder the terms: 5 + 2z = 4z2 Solving 5 + 2z = 4z2 Solving for variable 'z'. Combine like terms: 4z2 + -4z2 = 0 5 + 2z + -4z2 = 0 Begin completing the square. Divide all terms by -4 the coefficient of the squared term: Divide each side by '-4'. -1.25 + -0.5z + z2 = 0 Move the constant term to the right: Add '1.25' to each side of the equation. -1.25 + -0.5z + 1.25 + z2 = 0 + 1.25 Reorder the terms: -1.25 + 1.25 + -0.5z + z2 = 0 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + -0.5z + z2 = 0 + 1.25 -0.5z + z2 = 0 + 1.25 Combine like terms: 0 + 1.25 = 1.25 -0.5z + z2 = 1.25 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 = 1.25 + 0.0625 Reorder the terms: 0.0625 + -0.5z + z2 = 1.25 + 0.0625 Combine like terms: 1.25 + 0.0625 = 1.3125 0.0625 + -0.5z + z2 = 1.3125 Factor a perfect square on the left side: (z + -0.25)(z + -0.25) = 1.3125 Calculate the square root of the right side: 1.145643924 Break this problem into two subproblems by setting (z + -0.25) equal to 1.145643924 and -1.145643924.Subproblem 1
z + -0.25 = 1.145643924 Simplifying z + -0.25 = 1.145643924 Reorder the terms: -0.25 + z = 1.145643924 Solving -0.25 + z = 1.145643924 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 = 1.145643924 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + z = 1.145643924 + 0.25 z = 1.145643924 + 0.25 Combine like terms: 1.145643924 + 0.25 = 1.395643924 z = 1.395643924 Simplifying z = 1.395643924Subproblem 2
z + -0.25 = -1.145643924 Simplifying z + -0.25 = -1.145643924 Reorder the terms: -0.25 + z = -1.145643924 Solving -0.25 + z = -1.145643924 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 = -1.145643924 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + z = -1.145643924 + 0.25 z = -1.145643924 + 0.25 Combine like terms: -1.145643924 + 0.25 = -0.895643924 z = -0.895643924 Simplifying z = -0.895643924Solution
The solution to the problem is based on the solutions from the subproblems. z = {1.395643924, -0.895643924}
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