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Simplifying 2z2 = 3z + 4 Reorder the terms: 2z2 = 4 + 3z Solving 2z2 = 4 + 3z Solving for variable 'z'. Reorder the terms: -4 + -3z + 2z2 = 4 + 3z + -4 + -3z Reorder the terms: -4 + -3z + 2z2 = 4 + -4 + 3z + -3z Combine like terms: 4 + -4 = 0 -4 + -3z + 2z2 = 0 + 3z + -3z -4 + -3z + 2z2 = 3z + -3z Combine like terms: 3z + -3z = 0 -4 + -3z + 2z2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -2 + -1.5z + z2 = 0 Move the constant term to the right: Add '2' to each side of the equation. -2 + -1.5z + 2 + z2 = 0 + 2 Reorder the terms: -2 + 2 + -1.5z + z2 = 0 + 2 Combine like terms: -2 + 2 = 0 0 + -1.5z + z2 = 0 + 2 -1.5z + z2 = 0 + 2 Combine like terms: 0 + 2 = 2 -1.5z + z2 = 2 The z term is -1.5z. Take half its coefficient (-0.75). Square it (0.5625) and add it to both sides. Add '0.5625' to each side of the equation. -1.5z + 0.5625 + z2 = 2 + 0.5625 Reorder the terms: 0.5625 + -1.5z + z2 = 2 + 0.5625 Combine like terms: 2 + 0.5625 = 2.5625 0.5625 + -1.5z + z2 = 2.5625 Factor a perfect square on the left side: (z + -0.75)(z + -0.75) = 2.5625 Calculate the square root of the right side: 1.600781059 Break this problem into two subproblems by setting (z + -0.75) equal to 1.600781059 and -1.600781059.Subproblem 1
z + -0.75 = 1.600781059 Simplifying z + -0.75 = 1.600781059 Reorder the terms: -0.75 + z = 1.600781059 Solving -0.75 + z = 1.600781059 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + z = 1.600781059 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + z = 1.600781059 + 0.75 z = 1.600781059 + 0.75 Combine like terms: 1.600781059 + 0.75 = 2.350781059 z = 2.350781059 Simplifying z = 2.350781059Subproblem 2
z + -0.75 = -1.600781059 Simplifying z + -0.75 = -1.600781059 Reorder the terms: -0.75 + z = -1.600781059 Solving -0.75 + z = -1.600781059 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + z = -1.600781059 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + z = -1.600781059 + 0.75 z = -1.600781059 + 0.75 Combine like terms: -1.600781059 + 0.75 = -0.850781059 z = -0.850781059 Simplifying z = -0.850781059Solution
The solution to the problem is based on the solutions from the subproblems. z = {2.350781059, -0.850781059}
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