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Simplifying 4z2 + 31 = 24z Reorder the terms: 31 + 4z2 = 24z Solving 31 + 4z2 = 24z Solving for variable 'z'. Reorder the terms: 31 + -24z + 4z2 = 24z + -24z Combine like terms: 24z + -24z = 0 31 + -24z + 4z2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. 7.75 + -6z + z2 = 0 Move the constant term to the right: Add '-7.75' to each side of the equation. 7.75 + -6z + -7.75 + z2 = 0 + -7.75 Reorder the terms: 7.75 + -7.75 + -6z + z2 = 0 + -7.75 Combine like terms: 7.75 + -7.75 = 0.00 0.00 + -6z + z2 = 0 + -7.75 -6z + z2 = 0 + -7.75 Combine like terms: 0 + -7.75 = -7.75 -6z + z2 = -7.75 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 = -7.75 + 9 Reorder the terms: 9 + -6z + z2 = -7.75 + 9 Combine like terms: -7.75 + 9 = 1.25 9 + -6z + z2 = 1.25 Factor a perfect square on the left side: (z + -3)(z + -3) = 1.25 Calculate the square root of the right side: 1.118033989 Break this problem into two subproblems by setting (z + -3) equal to 1.118033989 and -1.118033989.Subproblem 1
z + -3 = 1.118033989 Simplifying z + -3 = 1.118033989 Reorder the terms: -3 + z = 1.118033989 Solving -3 + z = 1.118033989 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 = 1.118033989 + 3 Combine like terms: -3 + 3 = 0 0 + z = 1.118033989 + 3 z = 1.118033989 + 3 Combine like terms: 1.118033989 + 3 = 4.118033989 z = 4.118033989 Simplifying z = 4.118033989Subproblem 2
z + -3 = -1.118033989 Simplifying z + -3 = -1.118033989 Reorder the terms: -3 + z = -1.118033989 Solving -3 + z = -1.118033989 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 = -1.118033989 + 3 Combine like terms: -3 + 3 = 0 0 + z = -1.118033989 + 3 z = -1.118033989 + 3 Combine like terms: -1.118033989 + 3 = 1.881966011 z = 1.881966011 Simplifying z = 1.881966011Solution
The solution to the problem is based on the solutions from the subproblems. z = {4.118033989, 1.881966011}
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