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Simplifying (4 + 8) * 8 = (z + 12) * z Combine like terms: 4 + 8 = 12 (12) * 8 = (z + 12) * z Multiply 12 * 8 96 = (z + 12) * z Reorder the terms: 96 = (12 + z) * z Reorder the terms for easier multiplication: 96 = z(12 + z) 96 = (12 * z + z * z) 96 = (12z + z2) Solving 96 = 12z + z2 Solving for variable 'z'. Reorder the terms: 96 + -12z + -1z2 = 12z + -12z + z2 + -1z2 Combine like terms: 12z + -12z = 0 96 + -12z + -1z2 = 0 + z2 + -1z2 96 + -12z + -1z2 = z2 + -1z2 Combine like terms: z2 + -1z2 = 0 96 + -12z + -1z2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -96 + 12z + z2 = 0 Move the constant term to the right: Add '96' to each side of the equation. -96 + 12z + 96 + z2 = 0 + 96 Reorder the terms: -96 + 96 + 12z + z2 = 0 + 96 Combine like terms: -96 + 96 = 0 0 + 12z + z2 = 0 + 96 12z + z2 = 0 + 96 Combine like terms: 0 + 96 = 96 12z + z2 = 96 The z term is 12z. Take half its coefficient (6). Square it (36) and add it to both sides. Add '36' to each side of the equation. 12z + 36 + z2 = 96 + 36 Reorder the terms: 36 + 12z + z2 = 96 + 36 Combine like terms: 96 + 36 = 132 36 + 12z + z2 = 132 Factor a perfect square on the left side: (z + 6)(z + 6) = 132 Calculate the square root of the right side: 11.489125293 Break this problem into two subproblems by setting (z + 6) equal to 11.489125293 and -11.489125293.Subproblem 1
z + 6 = 11.489125293 Simplifying z + 6 = 11.489125293 Reorder the terms: 6 + z = 11.489125293 Solving 6 + z = 11.489125293 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-6' to each side of the equation. 6 + -6 + z = 11.489125293 + -6 Combine like terms: 6 + -6 = 0 0 + z = 11.489125293 + -6 z = 11.489125293 + -6 Combine like terms: 11.489125293 + -6 = 5.489125293 z = 5.489125293 Simplifying z = 5.489125293Subproblem 2
z + 6 = -11.489125293 Simplifying z + 6 = -11.489125293 Reorder the terms: 6 + z = -11.489125293 Solving 6 + z = -11.489125293 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-6' to each side of the equation. 6 + -6 + z = -11.489125293 + -6 Combine like terms: 6 + -6 = 0 0 + z = -11.489125293 + -6 z = -11.489125293 + -6 Combine like terms: -11.489125293 + -6 = -17.489125293 z = -17.489125293 Simplifying z = -17.489125293Solution
The solution to the problem is based on the solutions from the subproblems. z = {5.489125293, -17.489125293}
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