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Simplifying (z + 9)(5z + -2) = 18 Reorder the terms: (9 + z)(5z + -2) = 18 Reorder the terms: (9 + z)(-2 + 5z) = 18 Multiply (9 + z) * (-2 + 5z) (9(-2 + 5z) + z(-2 + 5z)) = 18 ((-2 * 9 + 5z * 9) + z(-2 + 5z)) = 18 ((-18 + 45z) + z(-2 + 5z)) = 18 (-18 + 45z + (-2 * z + 5z * z)) = 18 (-18 + 45z + (-2z + 5z2)) = 18 Combine like terms: 45z + -2z = 43z (-18 + 43z + 5z2) = 18 Solving -18 + 43z + 5z2 = 18 Solving for variable 'z'. Reorder the terms: -18 + -18 + 43z + 5z2 = 18 + -18 Combine like terms: -18 + -18 = -36 -36 + 43z + 5z2 = 18 + -18 Combine like terms: 18 + -18 = 0 -36 + 43z + 5z2 = 0 Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. -7.2 + 8.6z + z2 = 0 Move the constant term to the right: Add '7.2' to each side of the equation. -7.2 + 8.6z + 7.2 + z2 = 0 + 7.2 Reorder the terms: -7.2 + 7.2 + 8.6z + z2 = 0 + 7.2 Combine like terms: -7.2 + 7.2 = 0.0 0.0 + 8.6z + z2 = 0 + 7.2 8.6z + z2 = 0 + 7.2 Combine like terms: 0 + 7.2 = 7.2 8.6z + z2 = 7.2 The z term is 8.6z. Take half its coefficient (4.3). Square it (18.49) and add it to both sides. Add '18.49' to each side of the equation. 8.6z + 18.49 + z2 = 7.2 + 18.49 Reorder the terms: 18.49 + 8.6z + z2 = 7.2 + 18.49 Combine like terms: 7.2 + 18.49 = 25.69 18.49 + 8.6z + z2 = 25.69 Factor a perfect square on the left side: (z + 4.3)(z + 4.3) = 25.69 Calculate the square root of the right side: 5.068530359 Break this problem into two subproblems by setting (z + 4.3) equal to 5.068530359 and -5.068530359.Subproblem 1
z + 4.3 = 5.068530359 Simplifying z + 4.3 = 5.068530359 Reorder the terms: 4.3 + z = 5.068530359 Solving 4.3 + z = 5.068530359 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-4.3' to each side of the equation. 4.3 + -4.3 + z = 5.068530359 + -4.3 Combine like terms: 4.3 + -4.3 = 0.0 0.0 + z = 5.068530359 + -4.3 z = 5.068530359 + -4.3 Combine like terms: 5.068530359 + -4.3 = 0.768530359 z = 0.768530359 Simplifying z = 0.768530359Subproblem 2
z + 4.3 = -5.068530359 Simplifying z + 4.3 = -5.068530359 Reorder the terms: 4.3 + z = -5.068530359 Solving 4.3 + z = -5.068530359 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-4.3' to each side of the equation. 4.3 + -4.3 + z = -5.068530359 + -4.3 Combine like terms: 4.3 + -4.3 = 0.0 0.0 + z = -5.068530359 + -4.3 z = -5.068530359 + -4.3 Combine like terms: -5.068530359 + -4.3 = -9.368530359 z = -9.368530359 Simplifying z = -9.368530359Solution
The solution to the problem is based on the solutions from the subproblems. z = {0.768530359, -9.368530359}
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