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Simplifying (z + 6)(5z + -3) = 18 Reorder the terms: (6 + z)(5z + -3) = 18 Reorder the terms: (6 + z)(-3 + 5z) = 18 Multiply (6 + z) * (-3 + 5z) (6(-3 + 5z) + z(-3 + 5z)) = 18 ((-3 * 6 + 5z * 6) + z(-3 + 5z)) = 18 ((-18 + 30z) + z(-3 + 5z)) = 18 (-18 + 30z + (-3 * z + 5z * z)) = 18 (-18 + 30z + (-3z + 5z2)) = 18 Combine like terms: 30z + -3z = 27z (-18 + 27z + 5z2) = 18 Solving -18 + 27z + 5z2 = 18 Solving for variable 'z'. Reorder the terms: -18 + -18 + 27z + 5z2 = 18 + -18 Combine like terms: -18 + -18 = -36 -36 + 27z + 5z2 = 18 + -18 Combine like terms: 18 + -18 = 0 -36 + 27z + 5z2 = 0 Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. -7.2 + 5.4z + z2 = 0 Move the constant term to the right: Add '7.2' to each side of the equation. -7.2 + 5.4z + 7.2 + z2 = 0 + 7.2 Reorder the terms: -7.2 + 7.2 + 5.4z + z2 = 0 + 7.2 Combine like terms: -7.2 + 7.2 = 0.0 0.0 + 5.4z + z2 = 0 + 7.2 5.4z + z2 = 0 + 7.2 Combine like terms: 0 + 7.2 = 7.2 5.4z + z2 = 7.2 The z term is 5.4z. Take half its coefficient (2.7). Square it (7.29) and add it to both sides. Add '7.29' to each side of the equation. 5.4z + 7.29 + z2 = 7.2 + 7.29 Reorder the terms: 7.29 + 5.4z + z2 = 7.2 + 7.29 Combine like terms: 7.2 + 7.29 = 14.49 7.29 + 5.4z + z2 = 14.49 Factor a perfect square on the left side: (z + 2.7)(z + 2.7) = 14.49 Calculate the square root of the right side: 3.806573262 Break this problem into two subproblems by setting (z + 2.7) equal to 3.806573262 and -3.806573262.Subproblem 1
z + 2.7 = 3.806573262 Simplifying z + 2.7 = 3.806573262 Reorder the terms: 2.7 + z = 3.806573262 Solving 2.7 + z = 3.806573262 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-2.7' to each side of the equation. 2.7 + -2.7 + z = 3.806573262 + -2.7 Combine like terms: 2.7 + -2.7 = 0.0 0.0 + z = 3.806573262 + -2.7 z = 3.806573262 + -2.7 Combine like terms: 3.806573262 + -2.7 = 1.106573262 z = 1.106573262 Simplifying z = 1.106573262Subproblem 2
z + 2.7 = -3.806573262 Simplifying z + 2.7 = -3.806573262 Reorder the terms: 2.7 + z = -3.806573262 Solving 2.7 + z = -3.806573262 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-2.7' to each side of the equation. 2.7 + -2.7 + z = -3.806573262 + -2.7 Combine like terms: 2.7 + -2.7 = 0.0 0.0 + z = -3.806573262 + -2.7 z = -3.806573262 + -2.7 Combine like terms: -3.806573262 + -2.7 = -6.506573262 z = -6.506573262 Simplifying z = -6.506573262Solution
The solution to the problem is based on the solutions from the subproblems. z = {1.106573262, -6.506573262}
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