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Simplifying (6z + -5)(3z + -4) = 180 Reorder the terms: (-5 + 6z)(3z + -4) = 180 Reorder the terms: (-5 + 6z)(-4 + 3z) = 180 Multiply (-5 + 6z) * (-4 + 3z) (-5(-4 + 3z) + 6z * (-4 + 3z)) = 180 ((-4 * -5 + 3z * -5) + 6z * (-4 + 3z)) = 180 ((20 + -15z) + 6z * (-4 + 3z)) = 180 (20 + -15z + (-4 * 6z + 3z * 6z)) = 180 (20 + -15z + (-24z + 18z2)) = 180 Combine like terms: -15z + -24z = -39z (20 + -39z + 18z2) = 180 Solving 20 + -39z + 18z2 = 180 Solving for variable 'z'. Reorder the terms: 20 + -180 + -39z + 18z2 = 180 + -180 Combine like terms: 20 + -180 = -160 -160 + -39z + 18z2 = 180 + -180 Combine like terms: 180 + -180 = 0 -160 + -39z + 18z2 = 0 Begin completing the square. Divide all terms by 18 the coefficient of the squared term: Divide each side by '18'. -8.888888889 + -2.166666667z + z2 = 0 Move the constant term to the right: Add '8.888888889' to each side of the equation. -8.888888889 + -2.166666667z + 8.888888889 + z2 = 0 + 8.888888889 Reorder the terms: -8.888888889 + 8.888888889 + -2.166666667z + z2 = 0 + 8.888888889 Combine like terms: -8.888888889 + 8.888888889 = 0.000000000 0.000000000 + -2.166666667z + z2 = 0 + 8.888888889 -2.166666667z + z2 = 0 + 8.888888889 Combine like terms: 0 + 8.888888889 = 8.888888889 -2.166666667z + z2 = 8.888888889 The z term is -2.166666667z. Take half its coefficient (-1.083333334). Square it (1.173611113) and add it to both sides. Add '1.173611113' to each side of the equation. -2.166666667z + 1.173611113 + z2 = 8.888888889 + 1.173611113 Reorder the terms: 1.173611113 + -2.166666667z + z2 = 8.888888889 + 1.173611113 Combine like terms: 8.888888889 + 1.173611113 = 10.062500002 1.173611113 + -2.166666667z + z2 = 10.062500002 Factor a perfect square on the left side: (z + -1.083333334)(z + -1.083333334) = 10.062500002 Calculate the square root of the right side: 3.172144385 Break this problem into two subproblems by setting (z + -1.083333334) equal to 3.172144385 and -3.172144385.Subproblem 1
z + -1.083333334 = 3.172144385 Simplifying z + -1.083333334 = 3.172144385 Reorder the terms: -1.083333334 + z = 3.172144385 Solving -1.083333334 + z = 3.172144385 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '1.083333334' to each side of the equation. -1.083333334 + 1.083333334 + z = 3.172144385 + 1.083333334 Combine like terms: -1.083333334 + 1.083333334 = 0.000000000 0.000000000 + z = 3.172144385 + 1.083333334 z = 3.172144385 + 1.083333334 Combine like terms: 3.172144385 + 1.083333334 = 4.255477719 z = 4.255477719 Simplifying z = 4.255477719Subproblem 2
z + -1.083333334 = -3.172144385 Simplifying z + -1.083333334 = -3.172144385 Reorder the terms: -1.083333334 + z = -3.172144385 Solving -1.083333334 + z = -3.172144385 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '1.083333334' to each side of the equation. -1.083333334 + 1.083333334 + z = -3.172144385 + 1.083333334 Combine like terms: -1.083333334 + 1.083333334 = 0.000000000 0.000000000 + z = -3.172144385 + 1.083333334 z = -3.172144385 + 1.083333334 Combine like terms: -3.172144385 + 1.083333334 = -2.088811051 z = -2.088811051 Simplifying z = -2.088811051Solution
The solution to the problem is based on the solutions from the subproblems. z = {4.255477719, -2.088811051}
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