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Simplifying 21z2 + 85z + -21 = 0 Reorder the terms: -21 + 85z + 21z2 = 0 Solving -21 + 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 + 4.047619048z + z2 = 0 Move the constant term to the right: Add '1' to each side of the equation. -1 + 4.047619048z + 1 + z2 = 0 + 1 Reorder the terms: -1 + 1 + 4.047619048z + z2 = 0 + 1 Combine like terms: -1 + 1 = 0 0 + 4.047619048z + z2 = 0 + 1 4.047619048z + z2 = 0 + 1 Combine like terms: 0 + 1 = 1 4.047619048z + z2 = 1 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 + 4.095804989 Reorder the terms: 4.095804989 + 4.047619048z + z2 = 1 + 4.095804989 Combine like terms: 1 + 4.095804989 = 5.095804989 4.095804989 + 4.047619048z + z2 = 5.095804989 Factor a perfect square on the left side: (z + 2.023809524)(z + 2.023809524) = 5.095804989 Calculate the square root of the right side: 2.257388976 Break this problem into two subproblems by setting (z + 2.023809524) equal to 2.257388976 and -2.257388976.Subproblem 1
z + 2.023809524 = 2.257388976 Simplifying z + 2.023809524 = 2.257388976 Reorder the terms: 2.023809524 + z = 2.257388976 Solving 2.023809524 + z = 2.257388976 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 = 2.257388976 + -2.023809524 Combine like terms: 2.023809524 + -2.023809524 = 0.000000000 0.000000000 + z = 2.257388976 + -2.023809524 z = 2.257388976 + -2.023809524 Combine like terms: 2.257388976 + -2.023809524 = 0.233579452 z = 0.233579452 Simplifying z = 0.233579452Subproblem 2
z + 2.023809524 = -2.257388976 Simplifying z + 2.023809524 = -2.257388976 Reorder the terms: 2.023809524 + z = -2.257388976 Solving 2.023809524 + z = -2.257388976 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 = -2.257388976 + -2.023809524 Combine like terms: 2.023809524 + -2.023809524 = 0.000000000 0.000000000 + z = -2.257388976 + -2.023809524 z = -2.257388976 + -2.023809524 Combine like terms: -2.257388976 + -2.023809524 = -4.2811985 z = -4.2811985 Simplifying z = -4.2811985Solution
The solution to the problem is based on the solutions from the subproblems. z = {0.233579452, -4.2811985}
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