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Simplifying z2 + -12z = 32 Reorder the terms: -12z + z2 = 32 Solving -12z + z2 = 32 Solving for variable 'z'. Reorder the terms: -32 + -12z + z2 = 32 + -32 Combine like terms: 32 + -32 = 0 -32 + -12z + z2 = 0 Begin completing the square. Move the constant term to the right: Add '32' to each side of the equation. -32 + -12z + 32 + z2 = 0 + 32 Reorder the terms: -32 + 32 + -12z + z2 = 0 + 32 Combine like terms: -32 + 32 = 0 0 + -12z + z2 = 0 + 32 -12z + z2 = 0 + 32 Combine like terms: 0 + 32 = 32 -12z + z2 = 32 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 = 32 + 36 Reorder the terms: 36 + -12z + z2 = 32 + 36 Combine like terms: 32 + 36 = 68 36 + -12z + z2 = 68 Factor a perfect square on the left side: (z + -6)(z + -6) = 68 Calculate the square root of the right side: 8.246211251 Break this problem into two subproblems by setting (z + -6) equal to 8.246211251 and -8.246211251.Subproblem 1
z + -6 = 8.246211251 Simplifying z + -6 = 8.246211251 Reorder the terms: -6 + z = 8.246211251 Solving -6 + z = 8.246211251 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 = 8.246211251 + 6 Combine like terms: -6 + 6 = 0 0 + z = 8.246211251 + 6 z = 8.246211251 + 6 Combine like terms: 8.246211251 + 6 = 14.246211251 z = 14.246211251 Simplifying z = 14.246211251Subproblem 2
z + -6 = -8.246211251 Simplifying z + -6 = -8.246211251 Reorder the terms: -6 + z = -8.246211251 Solving -6 + z = -8.246211251 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 = -8.246211251 + 6 Combine like terms: -6 + 6 = 0 0 + z = -8.246211251 + 6 z = -8.246211251 + 6 Combine like terms: -8.246211251 + 6 = -2.246211251 z = -2.246211251 Simplifying z = -2.246211251Solution
The solution to the problem is based on the solutions from the subproblems. z = {14.246211251, -2.246211251}
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