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Simplifying 4z2 + 11 = 16z Reorder the terms: 11 + 4z2 = 16z Solving 11 + 4z2 = 16z Solving for variable 'z'. Reorder the terms: 11 + -16z + 4z2 = 16z + -16z Combine like terms: 16z + -16z = 0 11 + -16z + 4z2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. 2.75 + -4z + z2 = 0 Move the constant term to the right: Add '-2.75' to each side of the equation. 2.75 + -4z + -2.75 + z2 = 0 + -2.75 Reorder the terms: 2.75 + -2.75 + -4z + z2 = 0 + -2.75 Combine like terms: 2.75 + -2.75 = 0.00 0.00 + -4z + z2 = 0 + -2.75 -4z + z2 = 0 + -2.75 Combine like terms: 0 + -2.75 = -2.75 -4z + z2 = -2.75 The z term is -4z. Take half its coefficient (-2). Square it (4) and add it to both sides. Add '4' to each side of the equation. -4z + 4 + z2 = -2.75 + 4 Reorder the terms: 4 + -4z + z2 = -2.75 + 4 Combine like terms: -2.75 + 4 = 1.25 4 + -4z + z2 = 1.25 Factor a perfect square on the left side: (z + -2)(z + -2) = 1.25 Calculate the square root of the right side: 1.118033989 Break this problem into two subproblems by setting (z + -2) equal to 1.118033989 and -1.118033989.Subproblem 1
z + -2 = 1.118033989 Simplifying z + -2 = 1.118033989 Reorder the terms: -2 + z = 1.118033989 Solving -2 + z = 1.118033989 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + z = 1.118033989 + 2 Combine like terms: -2 + 2 = 0 0 + z = 1.118033989 + 2 z = 1.118033989 + 2 Combine like terms: 1.118033989 + 2 = 3.118033989 z = 3.118033989 Simplifying z = 3.118033989Subproblem 2
z + -2 = -1.118033989 Simplifying z + -2 = -1.118033989 Reorder the terms: -2 + z = -1.118033989 Solving -2 + z = -1.118033989 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + z = -1.118033989 + 2 Combine like terms: -2 + 2 = 0 0 + z = -1.118033989 + 2 z = -1.118033989 + 2 Combine like terms: -1.118033989 + 2 = 0.881966011 z = 0.881966011 Simplifying z = 0.881966011Solution
The solution to the problem is based on the solutions from the subproblems. z = {3.118033989, 0.881966011}
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