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Simplifying 13z2 + 49z + -9 = 0 Reorder the terms: -9 + 49z + 13z2 = 0 Solving -9 + 49z + 13z2 = 0 Solving for variable 'z'. Begin completing the square. Divide all terms by 13 the coefficient of the squared term: Divide each side by '13'. -0.6923076923 + 3.769230769z + z2 = 0 Move the constant term to the right: Add '0.6923076923' to each side of the equation. -0.6923076923 + 3.769230769z + 0.6923076923 + z2 = 0 + 0.6923076923 Reorder the terms: -0.6923076923 + 0.6923076923 + 3.769230769z + z2 = 0 + 0.6923076923 Combine like terms: -0.6923076923 + 0.6923076923 = 0.0000000000 0.0000000000 + 3.769230769z + z2 = 0 + 0.6923076923 3.769230769z + z2 = 0 + 0.6923076923 Combine like terms: 0 + 0.6923076923 = 0.6923076923 3.769230769z + z2 = 0.6923076923 The z term is 3.769230769z. Take half its coefficient (1.884615385). Square it (3.551775149) and add it to both sides. Add '3.551775149' to each side of the equation. 3.769230769z + 3.551775149 + z2 = 0.6923076923 + 3.551775149 Reorder the terms: 3.551775149 + 3.769230769z + z2 = 0.6923076923 + 3.551775149 Combine like terms: 0.6923076923 + 3.551775149 = 4.2440828413 3.551775149 + 3.769230769z + z2 = 4.2440828413 Factor a perfect square on the left side: (z + 1.884615385)(z + 1.884615385) = 4.2440828413 Calculate the square root of the right side: 2.060117191 Break this problem into two subproblems by setting (z + 1.884615385) equal to 2.060117191 and -2.060117191.Subproblem 1
z + 1.884615385 = 2.060117191 Simplifying z + 1.884615385 = 2.060117191 Reorder the terms: 1.884615385 + z = 2.060117191 Solving 1.884615385 + z = 2.060117191 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-1.884615385' to each side of the equation. 1.884615385 + -1.884615385 + z = 2.060117191 + -1.884615385 Combine like terms: 1.884615385 + -1.884615385 = 0.000000000 0.000000000 + z = 2.060117191 + -1.884615385 z = 2.060117191 + -1.884615385 Combine like terms: 2.060117191 + -1.884615385 = 0.175501806 z = 0.175501806 Simplifying z = 0.175501806Subproblem 2
z + 1.884615385 = -2.060117191 Simplifying z + 1.884615385 = -2.060117191 Reorder the terms: 1.884615385 + z = -2.060117191 Solving 1.884615385 + z = -2.060117191 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-1.884615385' to each side of the equation. 1.884615385 + -1.884615385 + z = -2.060117191 + -1.884615385 Combine like terms: 1.884615385 + -1.884615385 = 0.000000000 0.000000000 + z = -2.060117191 + -1.884615385 z = -2.060117191 + -1.884615385 Combine like terms: -2.060117191 + -1.884615385 = -3.944732576 z = -3.944732576 Simplifying z = -3.944732576Solution
The solution to the problem is based on the solutions from the subproblems. z = {0.175501806, -3.944732576}
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