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Simplifying z2 + 7z + -40 = 0 Reorder the terms: -40 + 7z + z2 = 0 Solving -40 + 7z + z2 = 0 Solving for variable 'z'. Begin completing the square. Move the constant term to the right: Add '40' to each side of the equation. -40 + 7z + 40 + z2 = 0 + 40 Reorder the terms: -40 + 40 + 7z + z2 = 0 + 40 Combine like terms: -40 + 40 = 0 0 + 7z + z2 = 0 + 40 7z + z2 = 0 + 40 Combine like terms: 0 + 40 = 40 7z + z2 = 40 The z term is 7z. Take half its coefficient (3.5). Square it (12.25) and add it to both sides. Add '12.25' to each side of the equation. 7z + 12.25 + z2 = 40 + 12.25 Reorder the terms: 12.25 + 7z + z2 = 40 + 12.25 Combine like terms: 40 + 12.25 = 52.25 12.25 + 7z + z2 = 52.25 Factor a perfect square on the left side: (z + 3.5)(z + 3.5) = 52.25 Calculate the square root of the right side: 7.228416147 Break this problem into two subproblems by setting (z + 3.5) equal to 7.228416147 and -7.228416147.Subproblem 1
z + 3.5 = 7.228416147 Simplifying z + 3.5 = 7.228416147 Reorder the terms: 3.5 + z = 7.228416147 Solving 3.5 + z = 7.228416147 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-3.5' to each side of the equation. 3.5 + -3.5 + z = 7.228416147 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + z = 7.228416147 + -3.5 z = 7.228416147 + -3.5 Combine like terms: 7.228416147 + -3.5 = 3.728416147 z = 3.728416147 Simplifying z = 3.728416147Subproblem 2
z + 3.5 = -7.228416147 Simplifying z + 3.5 = -7.228416147 Reorder the terms: 3.5 + z = -7.228416147 Solving 3.5 + z = -7.228416147 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-3.5' to each side of the equation. 3.5 + -3.5 + z = -7.228416147 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + z = -7.228416147 + -3.5 z = -7.228416147 + -3.5 Combine like terms: -7.228416147 + -3.5 = -10.728416147 z = -10.728416147 Simplifying z = -10.728416147Solution
The solution to the problem is based on the solutions from the subproblems. z = {3.728416147, -10.728416147}
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