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Simplifying z(6z + 30) = (z + -15) Reorder the terms: z(30 + 6z) = (z + -15) (30 * z + 6z * z) = (z + -15) (30z + 6z2) = (z + -15) Reorder the terms: 30z + 6z2 = (-15 + z) Remove parenthesis around (-15 + z) 30z + 6z2 = -15 + z Solving 30z + 6z2 = -15 + z Solving for variable 'z'. Reorder the terms: 15 + 30z + -1z + 6z2 = -15 + z + 15 + -1z Combine like terms: 30z + -1z = 29z 15 + 29z + 6z2 = -15 + z + 15 + -1z Reorder the terms: 15 + 29z + 6z2 = -15 + 15 + z + -1z Combine like terms: -15 + 15 = 0 15 + 29z + 6z2 = 0 + z + -1z 15 + 29z + 6z2 = z + -1z Combine like terms: z + -1z = 0 15 + 29z + 6z2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. 2.5 + 4.833333333z + z2 = 0 Move the constant term to the right: Add '-2.5' to each side of the equation. 2.5 + 4.833333333z + -2.5 + z2 = 0 + -2.5 Reorder the terms: 2.5 + -2.5 + 4.833333333z + z2 = 0 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + 4.833333333z + z2 = 0 + -2.5 4.833333333z + z2 = 0 + -2.5 Combine like terms: 0 + -2.5 = -2.5 4.833333333z + z2 = -2.5 The z term is 4.833333333z. Take half its coefficient (2.416666667). Square it (5.840277779) and add it to both sides. Add '5.840277779' to each side of the equation. 4.833333333z + 5.840277779 + z2 = -2.5 + 5.840277779 Reorder the terms: 5.840277779 + 4.833333333z + z2 = -2.5 + 5.840277779 Combine like terms: -2.5 + 5.840277779 = 3.340277779 5.840277779 + 4.833333333z + z2 = 3.340277779 Factor a perfect square on the left side: (z + 2.416666667)(z + 2.416666667) = 3.340277779 Calculate the square root of the right side: 1.827642684 Break this problem into two subproblems by setting (z + 2.416666667) equal to 1.827642684 and -1.827642684.Subproblem 1
z + 2.416666667 = 1.827642684 Simplifying z + 2.416666667 = 1.827642684 Reorder the terms: 2.416666667 + z = 1.827642684 Solving 2.416666667 + z = 1.827642684 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-2.416666667' to each side of the equation. 2.416666667 + -2.416666667 + z = 1.827642684 + -2.416666667 Combine like terms: 2.416666667 + -2.416666667 = 0.000000000 0.000000000 + z = 1.827642684 + -2.416666667 z = 1.827642684 + -2.416666667 Combine like terms: 1.827642684 + -2.416666667 = -0.589023983 z = -0.589023983 Simplifying z = -0.589023983Subproblem 2
z + 2.416666667 = -1.827642684 Simplifying z + 2.416666667 = -1.827642684 Reorder the terms: 2.416666667 + z = -1.827642684 Solving 2.416666667 + z = -1.827642684 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-2.416666667' to each side of the equation. 2.416666667 + -2.416666667 + z = -1.827642684 + -2.416666667 Combine like terms: 2.416666667 + -2.416666667 = 0.000000000 0.000000000 + z = -1.827642684 + -2.416666667 z = -1.827642684 + -2.416666667 Combine like terms: -1.827642684 + -2.416666667 = -4.244309351 z = -4.244309351 Simplifying z = -4.244309351Solution
The solution to the problem is based on the solutions from the subproblems. z = {-0.589023983, -4.244309351}
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