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Simplifying -2z(z + -4) + -1(3z + -2) = -2 + -1(6z + -2) Reorder the terms: -2z(-4 + z) + -1(3z + -2) = -2 + -1(6z + -2) (-4 * -2z + z * -2z) + -1(3z + -2) = -2 + -1(6z + -2) (8z + -2z2) + -1(3z + -2) = -2 + -1(6z + -2) Reorder the terms: 8z + -2z2 + -1(-2 + 3z) = -2 + -1(6z + -2) 8z + -2z2 + (-2 * -1 + 3z * -1) = -2 + -1(6z + -2) 8z + -2z2 + (2 + -3z) = -2 + -1(6z + -2) Reorder the terms: 2 + 8z + -3z + -2z2 = -2 + -1(6z + -2) Combine like terms: 8z + -3z = 5z 2 + 5z + -2z2 = -2 + -1(6z + -2) Reorder the terms: 2 + 5z + -2z2 = -2 + -1(-2 + 6z) 2 + 5z + -2z2 = -2 + (-2 * -1 + 6z * -1) 2 + 5z + -2z2 = -2 + (2 + -6z) Combine like terms: -2 + 2 = 0 2 + 5z + -2z2 = 0 + -6z 2 + 5z + -2z2 = -6z Solving 2 + 5z + -2z2 = -6z Solving for variable 'z'. Reorder the terms: 2 + 5z + 6z + -2z2 = -6z + 6z Combine like terms: 5z + 6z = 11z 2 + 11z + -2z2 = -6z + 6z Combine like terms: -6z + 6z = 0 2 + 11z + -2z2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -1 + -5.5z + z2 = 0 Move the constant term to the right: Add '1' to each side of the equation. -1 + -5.5z + 1 + z2 = 0 + 1 Reorder the terms: -1 + 1 + -5.5z + z2 = 0 + 1 Combine like terms: -1 + 1 = 0 0 + -5.5z + z2 = 0 + 1 -5.5z + z2 = 0 + 1 Combine like terms: 0 + 1 = 1 -5.5z + z2 = 1 The z term is -5.5z. Take half its coefficient (-2.75). Square it (7.5625) and add it to both sides. Add '7.5625' to each side of the equation. -5.5z + 7.5625 + z2 = 1 + 7.5625 Reorder the terms: 7.5625 + -5.5z + z2 = 1 + 7.5625 Combine like terms: 1 + 7.5625 = 8.5625 7.5625 + -5.5z + z2 = 8.5625 Factor a perfect square on the left side: (z + -2.75)(z + -2.75) = 8.5625 Calculate the square root of the right side: 2.926174978 Break this problem into two subproblems by setting (z + -2.75) equal to 2.926174978 and -2.926174978.Subproblem 1
z + -2.75 = 2.926174978 Simplifying z + -2.75 = 2.926174978 Reorder the terms: -2.75 + z = 2.926174978 Solving -2.75 + z = 2.926174978 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '2.75' to each side of the equation. -2.75 + 2.75 + z = 2.926174978 + 2.75 Combine like terms: -2.75 + 2.75 = 0.00 0.00 + z = 2.926174978 + 2.75 z = 2.926174978 + 2.75 Combine like terms: 2.926174978 + 2.75 = 5.676174978 z = 5.676174978 Simplifying z = 5.676174978Subproblem 2
z + -2.75 = -2.926174978 Simplifying z + -2.75 = -2.926174978 Reorder the terms: -2.75 + z = -2.926174978 Solving -2.75 + z = -2.926174978 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '2.75' to each side of the equation. -2.75 + 2.75 + z = -2.926174978 + 2.75 Combine like terms: -2.75 + 2.75 = 0.00 0.00 + z = -2.926174978 + 2.75 z = -2.926174978 + 2.75 Combine like terms: -2.926174978 + 2.75 = -0.176174978 z = -0.176174978 Simplifying z = -0.176174978Solution
The solution to the problem is based on the solutions from the subproblems. z = {5.676174978, -0.176174978}
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