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Simplifying z2 + -13z + 18 = 0 Reorder the terms: 18 + -13z + z2 = 0 Solving 18 + -13z + z2 = 0 Solving for variable 'z'. Begin completing the square. Move the constant term to the right: Add '-18' to each side of the equation. 18 + -13z + -18 + z2 = 0 + -18 Reorder the terms: 18 + -18 + -13z + z2 = 0 + -18 Combine like terms: 18 + -18 = 0 0 + -13z + z2 = 0 + -18 -13z + z2 = 0 + -18 Combine like terms: 0 + -18 = -18 -13z + z2 = -18 The z term is -13z. Take half its coefficient (-6.5). Square it (42.25) and add it to both sides. Add '42.25' to each side of the equation. -13z + 42.25 + z2 = -18 + 42.25 Reorder the terms: 42.25 + -13z + z2 = -18 + 42.25 Combine like terms: -18 + 42.25 = 24.25 42.25 + -13z + z2 = 24.25 Factor a perfect square on the left side: (z + -6.5)(z + -6.5) = 24.25 Calculate the square root of the right side: 4.924428901 Break this problem into two subproblems by setting (z + -6.5) equal to 4.924428901 and -4.924428901.Subproblem 1
z + -6.5 = 4.924428901 Simplifying z + -6.5 = 4.924428901 Reorder the terms: -6.5 + z = 4.924428901 Solving -6.5 + z = 4.924428901 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '6.5' to each side of the equation. -6.5 + 6.5 + z = 4.924428901 + 6.5 Combine like terms: -6.5 + 6.5 = 0.0 0.0 + z = 4.924428901 + 6.5 z = 4.924428901 + 6.5 Combine like terms: 4.924428901 + 6.5 = 11.424428901 z = 11.424428901 Simplifying z = 11.424428901Subproblem 2
z + -6.5 = -4.924428901 Simplifying z + -6.5 = -4.924428901 Reorder the terms: -6.5 + z = -4.924428901 Solving -6.5 + z = -4.924428901 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '6.5' to each side of the equation. -6.5 + 6.5 + z = -4.924428901 + 6.5 Combine like terms: -6.5 + 6.5 = 0.0 0.0 + z = -4.924428901 + 6.5 z = -4.924428901 + 6.5 Combine like terms: -4.924428901 + 6.5 = 1.575571099 z = 1.575571099 Simplifying z = 1.575571099Solution
The solution to the problem is based on the solutions from the subproblems. z = {11.424428901, 1.575571099}
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