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Simplifying z2 = 12 + 3z Solving z2 = 12 + 3z Solving for variable 'z'. Reorder the terms: -12 + -3z + z2 = 12 + 3z + -12 + -3z Reorder the terms: -12 + -3z + z2 = 12 + -12 + 3z + -3z Combine like terms: 12 + -12 = 0 -12 + -3z + z2 = 0 + 3z + -3z -12 + -3z + z2 = 3z + -3z Combine like terms: 3z + -3z = 0 -12 + -3z + z2 = 0 Begin completing the square. Move the constant term to the right: Add '12' to each side of the equation. -12 + -3z + 12 + z2 = 0 + 12 Reorder the terms: -12 + 12 + -3z + z2 = 0 + 12 Combine like terms: -12 + 12 = 0 0 + -3z + z2 = 0 + 12 -3z + z2 = 0 + 12 Combine like terms: 0 + 12 = 12 -3z + z2 = 12 The z term is -3z. Take half its coefficient (-1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. -3z + 2.25 + z2 = 12 + 2.25 Reorder the terms: 2.25 + -3z + z2 = 12 + 2.25 Combine like terms: 12 + 2.25 = 14.25 2.25 + -3z + z2 = 14.25 Factor a perfect square on the left side: (z + -1.5)(z + -1.5) = 14.25 Calculate the square root of the right side: 3.774917218 Break this problem into two subproblems by setting (z + -1.5) equal to 3.774917218 and -3.774917218.Subproblem 1
z + -1.5 = 3.774917218 Simplifying z + -1.5 = 3.774917218 Reorder the terms: -1.5 + z = 3.774917218 Solving -1.5 + z = 3.774917218 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + z = 3.774917218 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + z = 3.774917218 + 1.5 z = 3.774917218 + 1.5 Combine like terms: 3.774917218 + 1.5 = 5.274917218 z = 5.274917218 Simplifying z = 5.274917218Subproblem 2
z + -1.5 = -3.774917218 Simplifying z + -1.5 = -3.774917218 Reorder the terms: -1.5 + z = -3.774917218 Solving -1.5 + z = -3.774917218 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + z = -3.774917218 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + z = -3.774917218 + 1.5 z = -3.774917218 + 1.5 Combine like terms: -3.774917218 + 1.5 = -2.274917218 z = -2.274917218 Simplifying z = -2.274917218Solution
The solution to the problem is based on the solutions from the subproblems. z = {5.274917218, -2.274917218}
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