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Simplifying 5z2 + 19z + -5 = 0 Reorder the terms: -5 + 19z + 5z2 = 0 Solving -5 + 19z + 5z2 = 0 Solving for variable 'z'. Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. -1 + 3.8z + z2 = 0 Move the constant term to the right: Add '1' to each side of the equation. -1 + 3.8z + 1 + z2 = 0 + 1 Reorder the terms: -1 + 1 + 3.8z + z2 = 0 + 1 Combine like terms: -1 + 1 = 0 0 + 3.8z + z2 = 0 + 1 3.8z + z2 = 0 + 1 Combine like terms: 0 + 1 = 1 3.8z + z2 = 1 The z term is 3.8z. Take half its coefficient (1.9). Square it (3.61) and add it to both sides. Add '3.61' to each side of the equation. 3.8z + 3.61 + z2 = 1 + 3.61 Reorder the terms: 3.61 + 3.8z + z2 = 1 + 3.61 Combine like terms: 1 + 3.61 = 4.61 3.61 + 3.8z + z2 = 4.61 Factor a perfect square on the left side: (z + 1.9)(z + 1.9) = 4.61 Calculate the square root of the right side: 2.147091055 Break this problem into two subproblems by setting (z + 1.9) equal to 2.147091055 and -2.147091055.Subproblem 1
z + 1.9 = 2.147091055 Simplifying z + 1.9 = 2.147091055 Reorder the terms: 1.9 + z = 2.147091055 Solving 1.9 + z = 2.147091055 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-1.9' to each side of the equation. 1.9 + -1.9 + z = 2.147091055 + -1.9 Combine like terms: 1.9 + -1.9 = 0.0 0.0 + z = 2.147091055 + -1.9 z = 2.147091055 + -1.9 Combine like terms: 2.147091055 + -1.9 = 0.247091055 z = 0.247091055 Simplifying z = 0.247091055Subproblem 2
z + 1.9 = -2.147091055 Simplifying z + 1.9 = -2.147091055 Reorder the terms: 1.9 + z = -2.147091055 Solving 1.9 + z = -2.147091055 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-1.9' to each side of the equation. 1.9 + -1.9 + z = -2.147091055 + -1.9 Combine like terms: 1.9 + -1.9 = 0.0 0.0 + z = -2.147091055 + -1.9 z = -2.147091055 + -1.9 Combine like terms: -2.147091055 + -1.9 = -4.047091055 z = -4.047091055 Simplifying z = -4.047091055Solution
The solution to the problem is based on the solutions from the subproblems. z = {0.247091055, -4.047091055}
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