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Simplifying z2 + -16z = -19 Reorder the terms: -16z + z2 = -19 Solving -16z + z2 = -19 Solving for variable 'z'. Reorder the terms: 19 + -16z + z2 = -19 + 19 Combine like terms: -19 + 19 = 0 19 + -16z + z2 = 0 Begin completing the square. Move the constant term to the right: Add '-19' to each side of the equation. 19 + -16z + -19 + z2 = 0 + -19 Reorder the terms: 19 + -19 + -16z + z2 = 0 + -19 Combine like terms: 19 + -19 = 0 0 + -16z + z2 = 0 + -19 -16z + z2 = 0 + -19 Combine like terms: 0 + -19 = -19 -16z + z2 = -19 The z term is -16z. Take half its coefficient (-8). Square it (64) and add it to both sides. Add '64' to each side of the equation. -16z + 64 + z2 = -19 + 64 Reorder the terms: 64 + -16z + z2 = -19 + 64 Combine like terms: -19 + 64 = 45 64 + -16z + z2 = 45 Factor a perfect square on the left side: (z + -8)(z + -8) = 45 Calculate the square root of the right side: 6.708203933 Break this problem into two subproblems by setting (z + -8) equal to 6.708203933 and -6.708203933.Subproblem 1
z + -8 = 6.708203933 Simplifying z + -8 = 6.708203933 Reorder the terms: -8 + z = 6.708203933 Solving -8 + z = 6.708203933 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '8' to each side of the equation. -8 + 8 + z = 6.708203933 + 8 Combine like terms: -8 + 8 = 0 0 + z = 6.708203933 + 8 z = 6.708203933 + 8 Combine like terms: 6.708203933 + 8 = 14.708203933 z = 14.708203933 Simplifying z = 14.708203933Subproblem 2
z + -8 = -6.708203933 Simplifying z + -8 = -6.708203933 Reorder the terms: -8 + z = -6.708203933 Solving -8 + z = -6.708203933 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '8' to each side of the equation. -8 + 8 + z = -6.708203933 + 8 Combine like terms: -8 + 8 = 0 0 + z = -6.708203933 + 8 z = -6.708203933 + 8 Combine like terms: -6.708203933 + 8 = 1.291796067 z = 1.291796067 Simplifying z = 1.291796067Solution
The solution to the problem is based on the solutions from the subproblems. z = {14.708203933, 1.291796067}
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