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Simplifying (3z + 1)(2z + -50) = -5 Reorder the terms: (1 + 3z)(2z + -50) = -5 Reorder the terms: (1 + 3z)(-50 + 2z) = -5 Multiply (1 + 3z) * (-50 + 2z) (1(-50 + 2z) + 3z * (-50 + 2z)) = -5 ((-50 * 1 + 2z * 1) + 3z * (-50 + 2z)) = -5 ((-50 + 2z) + 3z * (-50 + 2z)) = -5 (-50 + 2z + (-50 * 3z + 2z * 3z)) = -5 (-50 + 2z + (-150z + 6z2)) = -5 Combine like terms: 2z + -150z = -148z (-50 + -148z + 6z2) = -5 Solving -50 + -148z + 6z2 = -5 Solving for variable 'z'. Reorder the terms: -50 + 5 + -148z + 6z2 = -5 + 5 Combine like terms: -50 + 5 = -45 -45 + -148z + 6z2 = -5 + 5 Combine like terms: -5 + 5 = 0 -45 + -148z + 6z2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -7.5 + -24.66666667z + z2 = 0 Move the constant term to the right: Add '7.5' to each side of the equation. -7.5 + -24.66666667z + 7.5 + z2 = 0 + 7.5 Reorder the terms: -7.5 + 7.5 + -24.66666667z + z2 = 0 + 7.5 Combine like terms: -7.5 + 7.5 = 0.0 0.0 + -24.66666667z + z2 = 0 + 7.5 -24.66666667z + z2 = 0 + 7.5 Combine like terms: 0 + 7.5 = 7.5 -24.66666667z + z2 = 7.5 The z term is -24.66666667z. Take half its coefficient (-12.33333334). Square it (152.1111113) and add it to both sides. Add '152.1111113' to each side of the equation. -24.66666667z + 152.1111113 + z2 = 7.5 + 152.1111113 Reorder the terms: 152.1111113 + -24.66666667z + z2 = 7.5 + 152.1111113 Combine like terms: 7.5 + 152.1111113 = 159.6111113 152.1111113 + -24.66666667z + z2 = 159.6111113 Factor a perfect square on the left side: (z + -12.33333334)(z + -12.33333334) = 159.6111113 Calculate the square root of the right side: 12.633729113 Break this problem into two subproblems by setting (z + -12.33333334) equal to 12.633729113 and -12.633729113.Subproblem 1
z + -12.33333334 = 12.633729113 Simplifying z + -12.33333334 = 12.633729113 Reorder the terms: -12.33333334 + z = 12.633729113 Solving -12.33333334 + z = 12.633729113 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '12.33333334' to each side of the equation. -12.33333334 + 12.33333334 + z = 12.633729113 + 12.33333334 Combine like terms: -12.33333334 + 12.33333334 = 0.00000000 0.00000000 + z = 12.633729113 + 12.33333334 z = 12.633729113 + 12.33333334 Combine like terms: 12.633729113 + 12.33333334 = 24.967062453 z = 24.967062453 Simplifying z = 24.967062453Subproblem 2
z + -12.33333334 = -12.633729113 Simplifying z + -12.33333334 = -12.633729113 Reorder the terms: -12.33333334 + z = -12.633729113 Solving -12.33333334 + z = -12.633729113 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '12.33333334' to each side of the equation. -12.33333334 + 12.33333334 + z = -12.633729113 + 12.33333334 Combine like terms: -12.33333334 + 12.33333334 = 0.00000000 0.00000000 + z = -12.633729113 + 12.33333334 z = -12.633729113 + 12.33333334 Combine like terms: -12.633729113 + 12.33333334 = -0.300395773 z = -0.300395773 Simplifying z = -0.300395773Solution
The solution to the problem is based on the solutions from the subproblems. z = {24.967062453, -0.300395773}
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