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Simplifying 4z2 + 9z + -5 = 0 Reorder the terms: -5 + 9z + 4z2 = 0 Solving -5 + 9z + 4z2 = 0 Solving for variable 'z'. Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -1.25 + 2.25z + z2 = 0 Move the constant term to the right: Add '1.25' to each side of the equation. -1.25 + 2.25z + 1.25 + z2 = 0 + 1.25 Reorder the terms: -1.25 + 1.25 + 2.25z + z2 = 0 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + 2.25z + z2 = 0 + 1.25 2.25z + z2 = 0 + 1.25 Combine like terms: 0 + 1.25 = 1.25 2.25z + z2 = 1.25 The z term is 2.25z. Take half its coefficient (1.125). Square it (1.265625) and add it to both sides. Add '1.265625' to each side of the equation. 2.25z + 1.265625 + z2 = 1.25 + 1.265625 Reorder the terms: 1.265625 + 2.25z + z2 = 1.25 + 1.265625 Combine like terms: 1.25 + 1.265625 = 2.515625 1.265625 + 2.25z + z2 = 2.515625 Factor a perfect square on the left side: (z + 1.125)(z + 1.125) = 2.515625 Calculate the square root of the right side: 1.586072193 Break this problem into two subproblems by setting (z + 1.125) equal to 1.586072193 and -1.586072193.Subproblem 1
z + 1.125 = 1.586072193 Simplifying z + 1.125 = 1.586072193 Reorder the terms: 1.125 + z = 1.586072193 Solving 1.125 + z = 1.586072193 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-1.125' to each side of the equation. 1.125 + -1.125 + z = 1.586072193 + -1.125 Combine like terms: 1.125 + -1.125 = 0.000 0.000 + z = 1.586072193 + -1.125 z = 1.586072193 + -1.125 Combine like terms: 1.586072193 + -1.125 = 0.461072193 z = 0.461072193 Simplifying z = 0.461072193Subproblem 2
z + 1.125 = -1.586072193 Simplifying z + 1.125 = -1.586072193 Reorder the terms: 1.125 + z = -1.586072193 Solving 1.125 + z = -1.586072193 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-1.125' to each side of the equation. 1.125 + -1.125 + z = -1.586072193 + -1.125 Combine like terms: 1.125 + -1.125 = 0.000 0.000 + z = -1.586072193 + -1.125 z = -1.586072193 + -1.125 Combine like terms: -1.586072193 + -1.125 = -2.711072193 z = -2.711072193 Simplifying z = -2.711072193Solution
The solution to the problem is based on the solutions from the subproblems. z = {0.461072193, -2.711072193}
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