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Simplifying 13z2 + 49z + -5 = 0 Reorder the terms: -5 + 49z + 13z2 = 0 Solving -5 + 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'. -0.3846153846 + 3.769230769z + z2 = 0 Move the constant term to the right: Add '0.3846153846' to each side of the equation. -0.3846153846 + 3.769230769z + 0.3846153846 + z2 = 0 + 0.3846153846 Reorder the terms: -0.3846153846 + 0.3846153846 + 3.769230769z + z2 = 0 + 0.3846153846 Combine like terms: -0.3846153846 + 0.3846153846 = 0.0000000000 0.0000000000 + 3.769230769z + z2 = 0 + 0.3846153846 3.769230769z + z2 = 0 + 0.3846153846 Combine like terms: 0 + 0.3846153846 = 0.3846153846 3.769230769z + z2 = 0.3846153846 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 = 0.3846153846 + 3.551775149 Reorder the terms: 3.551775149 + 3.769230769z + z2 = 0.3846153846 + 3.551775149 Combine like terms: 0.3846153846 + 3.551775149 = 3.9363905336 3.551775149 + 3.769230769z + z2 = 3.9363905336 Factor a perfect square on the left side: (z + 1.884615385)(z + 1.884615385) = 3.9363905336 Calculate the square root of the right side: 1.984033904 Break this problem into two subproblems by setting (z + 1.884615385) equal to 1.984033904 and -1.984033904.Subproblem 1
z + 1.884615385 = 1.984033904 Simplifying z + 1.884615385 = 1.984033904 Reorder the terms: 1.884615385 + z = 1.984033904 Solving 1.884615385 + z = 1.984033904 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 = 1.984033904 + -1.884615385 Combine like terms: 1.884615385 + -1.884615385 = 0.000000000 0.000000000 + z = 1.984033904 + -1.884615385 z = 1.984033904 + -1.884615385 Combine like terms: 1.984033904 + -1.884615385 = 0.099418519 z = 0.099418519 Simplifying z = 0.099418519Subproblem 2
z + 1.884615385 = -1.984033904 Simplifying z + 1.884615385 = -1.984033904 Reorder the terms: 1.884615385 + z = -1.984033904 Solving 1.884615385 + z = -1.984033904 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 = -1.984033904 + -1.884615385 Combine like terms: 1.884615385 + -1.884615385 = 0.000000000 0.000000000 + z = -1.984033904 + -1.884615385 z = -1.984033904 + -1.884615385 Combine like terms: -1.984033904 + -1.884615385 = -3.868649289 z = -3.868649289 Simplifying z = -3.868649289Solution
The solution to the problem is based on the solutions from the subproblems. z = {0.099418519, -3.868649289}
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