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Simplifying 21z2 + 85z + -28 = 0 Reorder the terms: -28 + 85z + 21z2 = 0 Solving -28 + 85z + 21z2 = 0 Solving for variable 'z'. Begin completing the square. Divide all terms by 21 the coefficient of the squared term: Divide each side by '21'. -1.333333333 + 4.047619048z + z2 = 0 Move the constant term to the right: Add '1.333333333' to each side of the equation. -1.333333333 + 4.047619048z + 1.333333333 + z2 = 0 + 1.333333333 Reorder the terms: -1.333333333 + 1.333333333 + 4.047619048z + z2 = 0 + 1.333333333 Combine like terms: -1.333333333 + 1.333333333 = 0.000000000 0.000000000 + 4.047619048z + z2 = 0 + 1.333333333 4.047619048z + z2 = 0 + 1.333333333 Combine like terms: 0 + 1.333333333 = 1.333333333 4.047619048z + z2 = 1.333333333 The z term is 4.047619048z. Take half its coefficient (2.023809524). Square it (4.095804989) and add it to both sides. Add '4.095804989' to each side of the equation. 4.047619048z + 4.095804989 + z2 = 1.333333333 + 4.095804989 Reorder the terms: 4.095804989 + 4.047619048z + z2 = 1.333333333 + 4.095804989 Combine like terms: 1.333333333 + 4.095804989 = 5.429138322 4.095804989 + 4.047619048z + z2 = 5.429138322 Factor a perfect square on the left side: (z + 2.023809524)(z + 2.023809524) = 5.429138322 Calculate the square root of the right side: 2.330051141 Break this problem into two subproblems by setting (z + 2.023809524) equal to 2.330051141 and -2.330051141.Subproblem 1
z + 2.023809524 = 2.330051141 Simplifying z + 2.023809524 = 2.330051141 Reorder the terms: 2.023809524 + z = 2.330051141 Solving 2.023809524 + z = 2.330051141 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-2.023809524' to each side of the equation. 2.023809524 + -2.023809524 + z = 2.330051141 + -2.023809524 Combine like terms: 2.023809524 + -2.023809524 = 0.000000000 0.000000000 + z = 2.330051141 + -2.023809524 z = 2.330051141 + -2.023809524 Combine like terms: 2.330051141 + -2.023809524 = 0.306241617 z = 0.306241617 Simplifying z = 0.306241617Subproblem 2
z + 2.023809524 = -2.330051141 Simplifying z + 2.023809524 = -2.330051141 Reorder the terms: 2.023809524 + z = -2.330051141 Solving 2.023809524 + z = -2.330051141 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-2.023809524' to each side of the equation. 2.023809524 + -2.023809524 + z = -2.330051141 + -2.023809524 Combine like terms: 2.023809524 + -2.023809524 = 0.000000000 0.000000000 + z = -2.330051141 + -2.023809524 z = -2.330051141 + -2.023809524 Combine like terms: -2.330051141 + -2.023809524 = -4.353860665 z = -4.353860665 Simplifying z = -4.353860665Solution
The solution to the problem is based on the solutions from the subproblems. z = {0.306241617, -4.353860665}
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