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Simplifying z2 + -14z = -19 Reorder the terms: -14z + z2 = -19 Solving -14z + z2 = -19 Solving for variable 'z'. Reorder the terms: 19 + -14z + z2 = -19 + 19 Combine like terms: -19 + 19 = 0 19 + -14z + z2 = 0 Begin completing the square. Move the constant term to the right: Add '-19' to each side of the equation. 19 + -14z + -19 + z2 = 0 + -19 Reorder the terms: 19 + -19 + -14z + z2 = 0 + -19 Combine like terms: 19 + -19 = 0 0 + -14z + z2 = 0 + -19 -14z + z2 = 0 + -19 Combine like terms: 0 + -19 = -19 -14z + z2 = -19 The z term is -14z. Take half its coefficient (-7). Square it (49) and add it to both sides. Add '49' to each side of the equation. -14z + 49 + z2 = -19 + 49 Reorder the terms: 49 + -14z + z2 = -19 + 49 Combine like terms: -19 + 49 = 30 49 + -14z + z2 = 30 Factor a perfect square on the left side: (z + -7)(z + -7) = 30 Calculate the square root of the right side: 5.477225575 Break this problem into two subproblems by setting (z + -7) equal to 5.477225575 and -5.477225575.Subproblem 1
z + -7 = 5.477225575 Simplifying z + -7 = 5.477225575 Reorder the terms: -7 + z = 5.477225575 Solving -7 + z = 5.477225575 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '7' to each side of the equation. -7 + 7 + z = 5.477225575 + 7 Combine like terms: -7 + 7 = 0 0 + z = 5.477225575 + 7 z = 5.477225575 + 7 Combine like terms: 5.477225575 + 7 = 12.477225575 z = 12.477225575 Simplifying z = 12.477225575Subproblem 2
z + -7 = -5.477225575 Simplifying z + -7 = -5.477225575 Reorder the terms: -7 + z = -5.477225575 Solving -7 + z = -5.477225575 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '7' to each side of the equation. -7 + 7 + z = -5.477225575 + 7 Combine like terms: -7 + 7 = 0 0 + z = -5.477225575 + 7 z = -5.477225575 + 7 Combine like terms: -5.477225575 + 7 = 1.522774425 z = 1.522774425 Simplifying z = 1.522774425Solution
The solution to the problem is based on the solutions from the subproblems. z = {12.477225575, 1.522774425}
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