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Simplifying z2 + -4z = 76 Reorder the terms: -4z + z2 = 76 Solving -4z + z2 = 76 Solving for variable 'z'. Reorder the terms: -76 + -4z + z2 = 76 + -76 Combine like terms: 76 + -76 = 0 -76 + -4z + z2 = 0 Begin completing the square. Move the constant term to the right: Add '76' to each side of the equation. -76 + -4z + 76 + z2 = 0 + 76 Reorder the terms: -76 + 76 + -4z + z2 = 0 + 76 Combine like terms: -76 + 76 = 0 0 + -4z + z2 = 0 + 76 -4z + z2 = 0 + 76 Combine like terms: 0 + 76 = 76 -4z + z2 = 76 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 = 76 + 4 Reorder the terms: 4 + -4z + z2 = 76 + 4 Combine like terms: 76 + 4 = 80 4 + -4z + z2 = 80 Factor a perfect square on the left side: (z + -2)(z + -2) = 80 Calculate the square root of the right side: 8.94427191 Break this problem into two subproblems by setting (z + -2) equal to 8.94427191 and -8.94427191.Subproblem 1
z + -2 = 8.94427191 Simplifying z + -2 = 8.94427191 Reorder the terms: -2 + z = 8.94427191 Solving -2 + z = 8.94427191 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 = 8.94427191 + 2 Combine like terms: -2 + 2 = 0 0 + z = 8.94427191 + 2 z = 8.94427191 + 2 Combine like terms: 8.94427191 + 2 = 10.94427191 z = 10.94427191 Simplifying z = 10.94427191Subproblem 2
z + -2 = -8.94427191 Simplifying z + -2 = -8.94427191 Reorder the terms: -2 + z = -8.94427191 Solving -2 + z = -8.94427191 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 = -8.94427191 + 2 Combine like terms: -2 + 2 = 0 0 + z = -8.94427191 + 2 z = -8.94427191 + 2 Combine like terms: -8.94427191 + 2 = -6.94427191 z = -6.94427191 Simplifying z = -6.94427191Solution
The solution to the problem is based on the solutions from the subproblems. z = {10.94427191, -6.94427191}
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