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Simplifying 3z2 + -18z + 26 = 0 Reorder the terms: 26 + -18z + 3z2 = 0 Solving 26 + -18z + 3z2 = 0 Solving for variable 'z'. Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. 8.666666667 + -6z + z2 = 0 Move the constant term to the right: Add '-8.666666667' to each side of the equation. 8.666666667 + -6z + -8.666666667 + z2 = 0 + -8.666666667 Reorder the terms: 8.666666667 + -8.666666667 + -6z + z2 = 0 + -8.666666667 Combine like terms: 8.666666667 + -8.666666667 = 0.000000000 0.000000000 + -6z + z2 = 0 + -8.666666667 -6z + z2 = 0 + -8.666666667 Combine like terms: 0 + -8.666666667 = -8.666666667 -6z + z2 = -8.666666667 The z term is -6z. Take half its coefficient (-3). Square it (9) and add it to both sides. Add '9' to each side of the equation. -6z + 9 + z2 = -8.666666667 + 9 Reorder the terms: 9 + -6z + z2 = -8.666666667 + 9 Combine like terms: -8.666666667 + 9 = 0.333333333 9 + -6z + z2 = 0.333333333 Factor a perfect square on the left side: (z + -3)(z + -3) = 0.333333333 Calculate the square root of the right side: 0.577350269 Break this problem into two subproblems by setting (z + -3) equal to 0.577350269 and -0.577350269.Subproblem 1
z + -3 = 0.577350269 Simplifying z + -3 = 0.577350269 Reorder the terms: -3 + z = 0.577350269 Solving -3 + z = 0.577350269 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + z = 0.577350269 + 3 Combine like terms: -3 + 3 = 0 0 + z = 0.577350269 + 3 z = 0.577350269 + 3 Combine like terms: 0.577350269 + 3 = 3.577350269 z = 3.577350269 Simplifying z = 3.577350269Subproblem 2
z + -3 = -0.577350269 Simplifying z + -3 = -0.577350269 Reorder the terms: -3 + z = -0.577350269 Solving -3 + z = -0.577350269 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + z = -0.577350269 + 3 Combine like terms: -3 + 3 = 0 0 + z = -0.577350269 + 3 z = -0.577350269 + 3 Combine like terms: -0.577350269 + 3 = 2.422649731 z = 2.422649731 Simplifying z = 2.422649731Solution
The solution to the problem is based on the solutions from the subproblems. z = {3.577350269, 2.422649731}
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