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Simplifying -50 = [4 + -1(9 + z)](3z) -50 = [4 + (9 * -1 + z * -1)](3z) -50 = [4 + (-9 + -1z)](3z) Combine like terms: 4 + -9 = -5 -50 = [-5 + -1z](3z) Remove parenthesis around (3z) -50 = [-5 + -1z] * 3z Reorder the terms for easier multiplication: -50 = 3z[-5 + -1z] -50 = [-5 * 3z + -1z * 3z] -50 = [-15z + -3z2] Solving -50 = -15z + -3z2 Solving for variable 'z'. Reorder the terms: -50 + 15z + 3z2 = -15z + 15z + -3z2 + 3z2 Combine like terms: -15z + 15z = 0 -50 + 15z + 3z2 = 0 + -3z2 + 3z2 -50 + 15z + 3z2 = -3z2 + 3z2 Combine like terms: -3z2 + 3z2 = 0 -50 + 15z + 3z2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -16.66666667 + 5z + z2 = 0 Move the constant term to the right: Add '16.66666667' to each side of the equation. -16.66666667 + 5z + 16.66666667 + z2 = 0 + 16.66666667 Reorder the terms: -16.66666667 + 16.66666667 + 5z + z2 = 0 + 16.66666667 Combine like terms: -16.66666667 + 16.66666667 = 0.00000000 0.00000000 + 5z + z2 = 0 + 16.66666667 5z + z2 = 0 + 16.66666667 Combine like terms: 0 + 16.66666667 = 16.66666667 5z + z2 = 16.66666667 The z term is 5z. Take half its coefficient (2.5). Square it (6.25) and add it to both sides. Add '6.25' to each side of the equation. 5z + 6.25 + z2 = 16.66666667 + 6.25 Reorder the terms: 6.25 + 5z + z2 = 16.66666667 + 6.25 Combine like terms: 16.66666667 + 6.25 = 22.91666667 6.25 + 5z + z2 = 22.91666667 Factor a perfect square on the left side: (z + 2.5)(z + 2.5) = 22.91666667 Calculate the square root of the right side: 4.787135539 Break this problem into two subproblems by setting (z + 2.5) equal to 4.787135539 and -4.787135539.Subproblem 1
z + 2.5 = 4.787135539 Simplifying z + 2.5 = 4.787135539 Reorder the terms: 2.5 + z = 4.787135539 Solving 2.5 + z = 4.787135539 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + z = 4.787135539 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + z = 4.787135539 + -2.5 z = 4.787135539 + -2.5 Combine like terms: 4.787135539 + -2.5 = 2.287135539 z = 2.287135539 Simplifying z = 2.287135539Subproblem 2
z + 2.5 = -4.787135539 Simplifying z + 2.5 = -4.787135539 Reorder the terms: 2.5 + z = -4.787135539 Solving 2.5 + z = -4.787135539 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + z = -4.787135539 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + z = -4.787135539 + -2.5 z = -4.787135539 + -2.5 Combine like terms: -4.787135539 + -2.5 = -7.287135539 z = -7.287135539 Simplifying z = -7.287135539Solution
The solution to the problem is based on the solutions from the subproblems. z = {2.287135539, -7.287135539}
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