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Simplifying -4z(3z + 5) + 9(z + 2) = 0 Reorder the terms: -4z(5 + 3z) + 9(z + 2) = 0 (5 * -4z + 3z * -4z) + 9(z + 2) = 0 (-20z + -12z2) + 9(z + 2) = 0 Reorder the terms: -20z + -12z2 + 9(2 + z) = 0 -20z + -12z2 + (2 * 9 + z * 9) = 0 -20z + -12z2 + (18 + 9z) = 0 Reorder the terms: 18 + -20z + 9z + -12z2 = 0 Combine like terms: -20z + 9z = -11z 18 + -11z + -12z2 = 0 Solving 18 + -11z + -12z2 = 0 Solving for variable 'z'. Begin completing the square. Divide all terms by -12 the coefficient of the squared term: Divide each side by '-12'. -1.5 + 0.9166666667z + z2 = 0 Move the constant term to the right: Add '1.5' to each side of the equation. -1.5 + 0.9166666667z + 1.5 + z2 = 0 + 1.5 Reorder the terms: -1.5 + 1.5 + 0.9166666667z + z2 = 0 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + 0.9166666667z + z2 = 0 + 1.5 0.9166666667z + z2 = 0 + 1.5 Combine like terms: 0 + 1.5 = 1.5 0.9166666667z + z2 = 1.5 The z term is 0.9166666667z. Take half its coefficient (0.4583333334). Square it (0.2100694445) and add it to both sides. Add '0.2100694445' to each side of the equation. 0.9166666667z + 0.2100694445 + z2 = 1.5 + 0.2100694445 Reorder the terms: 0.2100694445 + 0.9166666667z + z2 = 1.5 + 0.2100694445 Combine like terms: 1.5 + 0.2100694445 = 1.7100694445 0.2100694445 + 0.9166666667z + z2 = 1.7100694445 Factor a perfect square on the left side: (z + 0.4583333334)(z + 0.4583333334) = 1.7100694445 Calculate the square root of the right side: 1.307696236 Break this problem into two subproblems by setting (z + 0.4583333334) equal to 1.307696236 and -1.307696236.Subproblem 1
z + 0.4583333334 = 1.307696236 Simplifying z + 0.4583333334 = 1.307696236 Reorder the terms: 0.4583333334 + z = 1.307696236 Solving 0.4583333334 + z = 1.307696236 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.4583333334' to each side of the equation. 0.4583333334 + -0.4583333334 + z = 1.307696236 + -0.4583333334 Combine like terms: 0.4583333334 + -0.4583333334 = 0.0000000000 0.0000000000 + z = 1.307696236 + -0.4583333334 z = 1.307696236 + -0.4583333334 Combine like terms: 1.307696236 + -0.4583333334 = 0.8493629026 z = 0.8493629026 Simplifying z = 0.8493629026Subproblem 2
z + 0.4583333334 = -1.307696236 Simplifying z + 0.4583333334 = -1.307696236 Reorder the terms: 0.4583333334 + z = -1.307696236 Solving 0.4583333334 + z = -1.307696236 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.4583333334' to each side of the equation. 0.4583333334 + -0.4583333334 + z = -1.307696236 + -0.4583333334 Combine like terms: 0.4583333334 + -0.4583333334 = 0.0000000000 0.0000000000 + z = -1.307696236 + -0.4583333334 z = -1.307696236 + -0.4583333334 Combine like terms: -1.307696236 + -0.4583333334 = -1.7660295694 z = -1.7660295694 Simplifying z = -1.7660295694Solution
The solution to the problem is based on the solutions from the subproblems. z = {0.8493629026, -1.7660295694}
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