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Simplifying (z + -5)(z) = (0.5z + 7) * 2 Reorder the terms: (-5 + z)(z) = (0.5z + 7) * 2 Reorder the terms for easier multiplication: z(-5 + z) = (0.5z + 7) * 2 (-5 * z + z * z) = (0.5z + 7) * 2 (-5z + z2) = (0.5z + 7) * 2 Reorder the terms: -5z + z2 = (7 + 0.5z) * 2 Reorder the terms for easier multiplication: -5z + z2 = 2(7 + 0.5z) -5z + z2 = (7 * 2 + 0.5z * 2) -5z + z2 = (14 + 1z) Solving -5z + z2 = 14 + 1z Solving for variable 'z'. Reorder the terms: -14 + -5z + -1z + z2 = 14 + 1z + -14 + -1z Combine like terms: -5z + -1z = -6z -14 + -6z + z2 = 14 + 1z + -14 + -1z Reorder the terms: -14 + -6z + z2 = 14 + -14 + 1z + -1z Combine like terms: 14 + -14 = 0 -14 + -6z + z2 = 0 + 1z + -1z -14 + -6z + z2 = 1z + -1z Combine like terms: 1z + -1z = 0 -14 + -6z + z2 = 0 Begin completing the square. Move the constant term to the right: Add '14' to each side of the equation. -14 + -6z + 14 + z2 = 0 + 14 Reorder the terms: -14 + 14 + -6z + z2 = 0 + 14 Combine like terms: -14 + 14 = 0 0 + -6z + z2 = 0 + 14 -6z + z2 = 0 + 14 Combine like terms: 0 + 14 = 14 -6z + z2 = 14 The z term is -6z. Take half its coefficient (-3). Square it (9) and add it to both sides. Add '9' to each side of the equation. -6z + 9 + z2 = 14 + 9 Reorder the terms: 9 + -6z + z2 = 14 + 9 Combine like terms: 14 + 9 = 23 9 + -6z + z2 = 23 Factor a perfect square on the left side: (z + -3)(z + -3) = 23 Calculate the square root of the right side: 4.795831523 Break this problem into two subproblems by setting (z + -3) equal to 4.795831523 and -4.795831523.Subproblem 1
z + -3 = 4.795831523 Simplifying z + -3 = 4.795831523 Reorder the terms: -3 + z = 4.795831523 Solving -3 + z = 4.795831523 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + z = 4.795831523 + 3 Combine like terms: -3 + 3 = 0 0 + z = 4.795831523 + 3 z = 4.795831523 + 3 Combine like terms: 4.795831523 + 3 = 7.795831523 z = 7.795831523 Simplifying z = 7.795831523Subproblem 2
z + -3 = -4.795831523 Simplifying z + -3 = -4.795831523 Reorder the terms: -3 + z = -4.795831523 Solving -3 + z = -4.795831523 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + z = -4.795831523 + 3 Combine like terms: -3 + 3 = 0 0 + z = -4.795831523 + 3 z = -4.795831523 + 3 Combine like terms: -4.795831523 + 3 = -1.795831523 z = -1.795831523 Simplifying z = -1.795831523Solution
The solution to the problem is based on the solutions from the subproblems. z = {7.795831523, -1.795831523}
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