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Simplifying (z + 12) * z = (4 + 8) * 4 Reorder the terms: (12 + z) * z = (4 + 8) * 4 Reorder the terms for easier multiplication: z(12 + z) = (4 + 8) * 4 (12 * z + z * z) = (4 + 8) * 4 (12z + z2) = (4 + 8) * 4 Combine like terms: 4 + 8 = 12 12z + z2 = (12) * 4 Multiply 12 * 4 12z + z2 = 48 Solving 12z + z2 = 48 Solving for variable 'z'. Reorder the terms: -48 + 12z + z2 = 48 + -48 Combine like terms: 48 + -48 = 0 -48 + 12z + z2 = 0 Begin completing the square. Move the constant term to the right: Add '48' to each side of the equation. -48 + 12z + 48 + z2 = 0 + 48 Reorder the terms: -48 + 48 + 12z + z2 = 0 + 48 Combine like terms: -48 + 48 = 0 0 + 12z + z2 = 0 + 48 12z + z2 = 0 + 48 Combine like terms: 0 + 48 = 48 12z + z2 = 48 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 = 48 + 36 Reorder the terms: 36 + 12z + z2 = 48 + 36 Combine like terms: 48 + 36 = 84 36 + 12z + z2 = 84 Factor a perfect square on the left side: (z + 6)(z + 6) = 84 Calculate the square root of the right side: 9.16515139 Break this problem into two subproblems by setting (z + 6) equal to 9.16515139 and -9.16515139.Subproblem 1
z + 6 = 9.16515139 Simplifying z + 6 = 9.16515139 Reorder the terms: 6 + z = 9.16515139 Solving 6 + z = 9.16515139 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 = 9.16515139 + -6 Combine like terms: 6 + -6 = 0 0 + z = 9.16515139 + -6 z = 9.16515139 + -6 Combine like terms: 9.16515139 + -6 = 3.16515139 z = 3.16515139 Simplifying z = 3.16515139Subproblem 2
z + 6 = -9.16515139 Simplifying z + 6 = -9.16515139 Reorder the terms: 6 + z = -9.16515139 Solving 6 + z = -9.16515139 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 = -9.16515139 + -6 Combine like terms: 6 + -6 = 0 0 + z = -9.16515139 + -6 z = -9.16515139 + -6 Combine like terms: -9.16515139 + -6 = -15.16515139 z = -15.16515139 Simplifying z = -15.16515139Solution
The solution to the problem is based on the solutions from the subproblems. z = {3.16515139, -15.16515139}
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