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Simplifying (z + 7)(5z + -4) = 18 Reorder the terms: (7 + z)(5z + -4) = 18 Reorder the terms: (7 + z)(-4 + 5z) = 18 Multiply (7 + z) * (-4 + 5z) (7(-4 + 5z) + z(-4 + 5z)) = 18 ((-4 * 7 + 5z * 7) + z(-4 + 5z)) = 18 ((-28 + 35z) + z(-4 + 5z)) = 18 (-28 + 35z + (-4 * z + 5z * z)) = 18 (-28 + 35z + (-4z + 5z2)) = 18 Combine like terms: 35z + -4z = 31z (-28 + 31z + 5z2) = 18 Solving -28 + 31z + 5z2 = 18 Solving for variable 'z'. Reorder the terms: -28 + -18 + 31z + 5z2 = 18 + -18 Combine like terms: -28 + -18 = -46 -46 + 31z + 5z2 = 18 + -18 Combine like terms: 18 + -18 = 0 -46 + 31z + 5z2 = 0 Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. -9.2 + 6.2z + z2 = 0 Move the constant term to the right: Add '9.2' to each side of the equation. -9.2 + 6.2z + 9.2 + z2 = 0 + 9.2 Reorder the terms: -9.2 + 9.2 + 6.2z + z2 = 0 + 9.2 Combine like terms: -9.2 + 9.2 = 0.0 0.0 + 6.2z + z2 = 0 + 9.2 6.2z + z2 = 0 + 9.2 Combine like terms: 0 + 9.2 = 9.2 6.2z + z2 = 9.2 The z term is 6.2z. Take half its coefficient (3.1). Square it (9.61) and add it to both sides. Add '9.61' to each side of the equation. 6.2z + 9.61 + z2 = 9.2 + 9.61 Reorder the terms: 9.61 + 6.2z + z2 = 9.2 + 9.61 Combine like terms: 9.2 + 9.61 = 18.81 9.61 + 6.2z + z2 = 18.81 Factor a perfect square on the left side: (z + 3.1)(z + 3.1) = 18.81 Calculate the square root of the right side: 4.337049688 Break this problem into two subproblems by setting (z + 3.1) equal to 4.337049688 and -4.337049688.Subproblem 1
z + 3.1 = 4.337049688 Simplifying z + 3.1 = 4.337049688 Reorder the terms: 3.1 + z = 4.337049688 Solving 3.1 + z = 4.337049688 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-3.1' to each side of the equation. 3.1 + -3.1 + z = 4.337049688 + -3.1 Combine like terms: 3.1 + -3.1 = 0.0 0.0 + z = 4.337049688 + -3.1 z = 4.337049688 + -3.1 Combine like terms: 4.337049688 + -3.1 = 1.237049688 z = 1.237049688 Simplifying z = 1.237049688Subproblem 2
z + 3.1 = -4.337049688 Simplifying z + 3.1 = -4.337049688 Reorder the terms: 3.1 + z = -4.337049688 Solving 3.1 + z = -4.337049688 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-3.1' to each side of the equation. 3.1 + -3.1 + z = -4.337049688 + -3.1 Combine like terms: 3.1 + -3.1 = 0.0 0.0 + z = -4.337049688 + -3.1 z = -4.337049688 + -3.1 Combine like terms: -4.337049688 + -3.1 = -7.437049688 z = -7.437049688 Simplifying z = -7.437049688Solution
The solution to the problem is based on the solutions from the subproblems. z = {1.237049688, -7.437049688}
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