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Simplifying z(2z + -3) = 17 Reorder the terms: z(-3 + 2z) = 17 (-3 * z + 2z * z) = 17 (-3z + 2z2) = 17 Solving -3z + 2z2 = 17 Solving for variable 'z'. Reorder the terms: -17 + -3z + 2z2 = 17 + -17 Combine like terms: 17 + -17 = 0 -17 + -3z + 2z2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -8.5 + -1.5z + z2 = 0 Move the constant term to the right: Add '8.5' to each side of the equation. -8.5 + -1.5z + 8.5 + z2 = 0 + 8.5 Reorder the terms: -8.5 + 8.5 + -1.5z + z2 = 0 + 8.5 Combine like terms: -8.5 + 8.5 = 0.0 0.0 + -1.5z + z2 = 0 + 8.5 -1.5z + z2 = 0 + 8.5 Combine like terms: 0 + 8.5 = 8.5 -1.5z + z2 = 8.5 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 = 8.5 + 0.5625 Reorder the terms: 0.5625 + -1.5z + z2 = 8.5 + 0.5625 Combine like terms: 8.5 + 0.5625 = 9.0625 0.5625 + -1.5z + z2 = 9.0625 Factor a perfect square on the left side: (z + -0.75)(z + -0.75) = 9.0625 Calculate the square root of the right side: 3.010398645 Break this problem into two subproblems by setting (z + -0.75) equal to 3.010398645 and -3.010398645.Subproblem 1
z + -0.75 = 3.010398645 Simplifying z + -0.75 = 3.010398645 Reorder the terms: -0.75 + z = 3.010398645 Solving -0.75 + z = 3.010398645 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 = 3.010398645 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + z = 3.010398645 + 0.75 z = 3.010398645 + 0.75 Combine like terms: 3.010398645 + 0.75 = 3.760398645 z = 3.760398645 Simplifying z = 3.760398645Subproblem 2
z + -0.75 = -3.010398645 Simplifying z + -0.75 = -3.010398645 Reorder the terms: -0.75 + z = -3.010398645 Solving -0.75 + z = -3.010398645 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 = -3.010398645 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + z = -3.010398645 + 0.75 z = -3.010398645 + 0.75 Combine like terms: -3.010398645 + 0.75 = -2.260398645 z = -2.260398645 Simplifying z = -2.260398645Solution
The solution to the problem is based on the solutions from the subproblems. z = {3.760398645, -2.260398645}
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