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Simplifying z2 + -13z = 20 Reorder the terms: -13z + z2 = 20 Solving -13z + z2 = 20 Solving for variable 'z'. Reorder the terms: -20 + -13z + z2 = 20 + -20 Combine like terms: 20 + -20 = 0 -20 + -13z + z2 = 0 Begin completing the square. Move the constant term to the right: Add '20' to each side of the equation. -20 + -13z + 20 + z2 = 0 + 20 Reorder the terms: -20 + 20 + -13z + z2 = 0 + 20 Combine like terms: -20 + 20 = 0 0 + -13z + z2 = 0 + 20 -13z + z2 = 0 + 20 Combine like terms: 0 + 20 = 20 -13z + z2 = 20 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 = 20 + 42.25 Reorder the terms: 42.25 + -13z + z2 = 20 + 42.25 Combine like terms: 20 + 42.25 = 62.25 42.25 + -13z + z2 = 62.25 Factor a perfect square on the left side: (z + -6.5)(z + -6.5) = 62.25 Calculate the square root of the right side: 7.889866919 Break this problem into two subproblems by setting (z + -6.5) equal to 7.889866919 and -7.889866919.Subproblem 1
z + -6.5 = 7.889866919 Simplifying z + -6.5 = 7.889866919 Reorder the terms: -6.5 + z = 7.889866919 Solving -6.5 + z = 7.889866919 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 = 7.889866919 + 6.5 Combine like terms: -6.5 + 6.5 = 0.0 0.0 + z = 7.889866919 + 6.5 z = 7.889866919 + 6.5 Combine like terms: 7.889866919 + 6.5 = 14.389866919 z = 14.389866919 Simplifying z = 14.389866919Subproblem 2
z + -6.5 = -7.889866919 Simplifying z + -6.5 = -7.889866919 Reorder the terms: -6.5 + z = -7.889866919 Solving -6.5 + z = -7.889866919 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 = -7.889866919 + 6.5 Combine like terms: -6.5 + 6.5 = 0.0 0.0 + z = -7.889866919 + 6.5 z = -7.889866919 + 6.5 Combine like terms: -7.889866919 + 6.5 = -1.389866919 z = -1.389866919 Simplifying z = -1.389866919Solution
The solution to the problem is based on the solutions from the subproblems. z = {14.389866919, -1.389866919}
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