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Simplifying z2 + 4z = 3 Reorder the terms: 4z + z2 = 3 Solving 4z + z2 = 3 Solving for variable 'z'. Reorder the terms: -3 + 4z + z2 = 3 + -3 Combine like terms: 3 + -3 = 0 -3 + 4z + z2 = 0 Begin completing the square. Move the constant term to the right: Add '3' to each side of the equation. -3 + 4z + 3 + z2 = 0 + 3 Reorder the terms: -3 + 3 + 4z + z2 = 0 + 3 Combine like terms: -3 + 3 = 0 0 + 4z + z2 = 0 + 3 4z + z2 = 0 + 3 Combine like terms: 0 + 3 = 3 4z + z2 = 3 The z term is 4z. Take half its coefficient (2). Square it (4) and add it to both sides. Add '4' to each side of the equation. 4z + 4 + z2 = 3 + 4 Reorder the terms: 4 + 4z + z2 = 3 + 4 Combine like terms: 3 + 4 = 7 4 + 4z + z2 = 7 Factor a perfect square on the left side: (z + 2)(z + 2) = 7 Calculate the square root of the right side: 2.645751311 Break this problem into two subproblems by setting (z + 2) equal to 2.645751311 and -2.645751311.Subproblem 1
z + 2 = 2.645751311 Simplifying z + 2 = 2.645751311 Reorder the terms: 2 + z = 2.645751311 Solving 2 + z = 2.645751311 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-2' to each side of the equation. 2 + -2 + z = 2.645751311 + -2 Combine like terms: 2 + -2 = 0 0 + z = 2.645751311 + -2 z = 2.645751311 + -2 Combine like terms: 2.645751311 + -2 = 0.645751311 z = 0.645751311 Simplifying z = 0.645751311Subproblem 2
z + 2 = -2.645751311 Simplifying z + 2 = -2.645751311 Reorder the terms: 2 + z = -2.645751311 Solving 2 + z = -2.645751311 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-2' to each side of the equation. 2 + -2 + z = -2.645751311 + -2 Combine like terms: 2 + -2 = 0 0 + z = -2.645751311 + -2 z = -2.645751311 + -2 Combine like terms: -2.645751311 + -2 = -4.645751311 z = -4.645751311 Simplifying z = -4.645751311Solution
The solution to the problem is based on the solutions from the subproblems. z = {0.645751311, -4.645751311}
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