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Simplifying 3z2 + -3z + -1 = 2z2 + -1z + 3 Reorder the terms: -1 + -3z + 3z2 = 2z2 + -1z + 3 Reorder the terms: -1 + -3z + 3z2 = 3 + -1z + 2z2 Solving -1 + -3z + 3z2 = 3 + -1z + 2z2 Solving for variable 'z'. Reorder the terms: -1 + -3 + -3z + z + 3z2 + -2z2 = 3 + -1z + 2z2 + -3 + z + -2z2 Combine like terms: -1 + -3 = -4 -4 + -3z + z + 3z2 + -2z2 = 3 + -1z + 2z2 + -3 + z + -2z2 Combine like terms: -3z + z = -2z -4 + -2z + 3z2 + -2z2 = 3 + -1z + 2z2 + -3 + z + -2z2 Combine like terms: 3z2 + -2z2 = 1z2 -4 + -2z + 1z2 = 3 + -1z + 2z2 + -3 + z + -2z2 Reorder the terms: -4 + -2z + 1z2 = 3 + -3 + -1z + z + 2z2 + -2z2 Combine like terms: 3 + -3 = 0 -4 + -2z + 1z2 = 0 + -1z + z + 2z2 + -2z2 -4 + -2z + 1z2 = -1z + z + 2z2 + -2z2 Combine like terms: -1z + z = 0 -4 + -2z + 1z2 = 0 + 2z2 + -2z2 -4 + -2z + 1z2 = 2z2 + -2z2 Combine like terms: 2z2 + -2z2 = 0 -4 + -2z + 1z2 = 0 Begin completing the square. Move the constant term to the right: Add '4' to each side of the equation. -4 + -2z + 4 + z2 = 0 + 4 Reorder the terms: -4 + 4 + -2z + z2 = 0 + 4 Combine like terms: -4 + 4 = 0 0 + -2z + z2 = 0 + 4 -2z + z2 = 0 + 4 Combine like terms: 0 + 4 = 4 -2z + z2 = 4 The z term is -2z. Take half its coefficient (-1). Square it (1) and add it to both sides. Add '1' to each side of the equation. -2z + 1 + z2 = 4 + 1 Reorder the terms: 1 + -2z + z2 = 4 + 1 Combine like terms: 4 + 1 = 5 1 + -2z + z2 = 5 Factor a perfect square on the left side: (z + -1)(z + -1) = 5 Calculate the square root of the right side: 2.236067978 Break this problem into two subproblems by setting (z + -1) equal to 2.236067978 and -2.236067978.Subproblem 1
z + -1 = 2.236067978 Simplifying z + -1 = 2.236067978 Reorder the terms: -1 + z = 2.236067978 Solving -1 + z = 2.236067978 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + z = 2.236067978 + 1 Combine like terms: -1 + 1 = 0 0 + z = 2.236067978 + 1 z = 2.236067978 + 1 Combine like terms: 2.236067978 + 1 = 3.236067978 z = 3.236067978 Simplifying z = 3.236067978Subproblem 2
z + -1 = -2.236067978 Simplifying z + -1 = -2.236067978 Reorder the terms: -1 + z = -2.236067978 Solving -1 + z = -2.236067978 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + z = -2.236067978 + 1 Combine like terms: -1 + 1 = 0 0 + z = -2.236067978 + 1 z = -2.236067978 + 1 Combine like terms: -2.236067978 + 1 = -1.236067978 z = -1.236067978 Simplifying z = -1.236067978Solution
The solution to the problem is based on the solutions from the subproblems. z = {3.236067978, -1.236067978}
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