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Simplifying 6z2 + 17z + -12 = 0 Reorder the terms: -12 + 17z + 6z2 = 0 Solving -12 + 17z + 6z2 = 0 Solving for variable 'z'. Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -2 + 2.833333333z + z2 = 0 Move the constant term to the right: Add '2' to each side of the equation. -2 + 2.833333333z + 2 + z2 = 0 + 2 Reorder the terms: -2 + 2 + 2.833333333z + z2 = 0 + 2 Combine like terms: -2 + 2 = 0 0 + 2.833333333z + z2 = 0 + 2 2.833333333z + z2 = 0 + 2 Combine like terms: 0 + 2 = 2 2.833333333z + z2 = 2 The z term is 2.833333333z. Take half its coefficient (1.416666667). Square it (2.006944445) and add it to both sides. Add '2.006944445' to each side of the equation. 2.833333333z + 2.006944445 + z2 = 2 + 2.006944445 Reorder the terms: 2.006944445 + 2.833333333z + z2 = 2 + 2.006944445 Combine like terms: 2 + 2.006944445 = 4.006944445 2.006944445 + 2.833333333z + z2 = 4.006944445 Factor a perfect square on the left side: (z + 1.416666667)(z + 1.416666667) = 4.006944445 Calculate the square root of the right side: 2.001735358 Break this problem into two subproblems by setting (z + 1.416666667) equal to 2.001735358 and -2.001735358.Subproblem 1
z + 1.416666667 = 2.001735358 Simplifying z + 1.416666667 = 2.001735358 Reorder the terms: 1.416666667 + z = 2.001735358 Solving 1.416666667 + z = 2.001735358 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-1.416666667' to each side of the equation. 1.416666667 + -1.416666667 + z = 2.001735358 + -1.416666667 Combine like terms: 1.416666667 + -1.416666667 = 0.000000000 0.000000000 + z = 2.001735358 + -1.416666667 z = 2.001735358 + -1.416666667 Combine like terms: 2.001735358 + -1.416666667 = 0.585068691 z = 0.585068691 Simplifying z = 0.585068691Subproblem 2
z + 1.416666667 = -2.001735358 Simplifying z + 1.416666667 = -2.001735358 Reorder the terms: 1.416666667 + z = -2.001735358 Solving 1.416666667 + z = -2.001735358 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-1.416666667' to each side of the equation. 1.416666667 + -1.416666667 + z = -2.001735358 + -1.416666667 Combine like terms: 1.416666667 + -1.416666667 = 0.000000000 0.000000000 + z = -2.001735358 + -1.416666667 z = -2.001735358 + -1.416666667 Combine like terms: -2.001735358 + -1.416666667 = -3.418402025 z = -3.418402025 Simplifying z = -3.418402025Solution
The solution to the problem is based on the solutions from the subproblems. z = {0.585068691, -3.418402025}
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