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Simplifying 13z2 + 6z = 35 Reorder the terms: 6z + 13z2 = 35 Solving 6z + 13z2 = 35 Solving for variable 'z'. Reorder the terms: -35 + 6z + 13z2 = 35 + -35 Combine like terms: 35 + -35 = 0 -35 + 6z + 13z2 = 0 Begin completing the square. Divide all terms by 13 the coefficient of the squared term: Divide each side by '13'. -2.692307692 + 0.4615384615z + z2 = 0 Move the constant term to the right: Add '2.692307692' to each side of the equation. -2.692307692 + 0.4615384615z + 2.692307692 + z2 = 0 + 2.692307692 Reorder the terms: -2.692307692 + 2.692307692 + 0.4615384615z + z2 = 0 + 2.692307692 Combine like terms: -2.692307692 + 2.692307692 = 0.000000000 0.000000000 + 0.4615384615z + z2 = 0 + 2.692307692 0.4615384615z + z2 = 0 + 2.692307692 Combine like terms: 0 + 2.692307692 = 2.692307692 0.4615384615z + z2 = 2.692307692 The z term is 0.4615384615z. Take half its coefficient (0.2307692308). Square it (0.05325443788) and add it to both sides. Add '0.05325443788' to each side of the equation. 0.4615384615z + 0.05325443788 + z2 = 2.692307692 + 0.05325443788 Reorder the terms: 0.05325443788 + 0.4615384615z + z2 = 2.692307692 + 0.05325443788 Combine like terms: 2.692307692 + 0.05325443788 = 2.74556212988 0.05325443788 + 0.4615384615z + z2 = 2.74556212988 Factor a perfect square on the left side: (z + 0.2307692308)(z + 0.2307692308) = 2.74556212988 Calculate the square root of the right side: 1.656973787 Break this problem into two subproblems by setting (z + 0.2307692308) equal to 1.656973787 and -1.656973787.Subproblem 1
z + 0.2307692308 = 1.656973787 Simplifying z + 0.2307692308 = 1.656973787 Reorder the terms: 0.2307692308 + z = 1.656973787 Solving 0.2307692308 + z = 1.656973787 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.2307692308' to each side of the equation. 0.2307692308 + -0.2307692308 + z = 1.656973787 + -0.2307692308 Combine like terms: 0.2307692308 + -0.2307692308 = 0.0000000000 0.0000000000 + z = 1.656973787 + -0.2307692308 z = 1.656973787 + -0.2307692308 Combine like terms: 1.656973787 + -0.2307692308 = 1.4262045562 z = 1.4262045562 Simplifying z = 1.4262045562Subproblem 2
z + 0.2307692308 = -1.656973787 Simplifying z + 0.2307692308 = -1.656973787 Reorder the terms: 0.2307692308 + z = -1.656973787 Solving 0.2307692308 + z = -1.656973787 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.2307692308' to each side of the equation. 0.2307692308 + -0.2307692308 + z = -1.656973787 + -0.2307692308 Combine like terms: 0.2307692308 + -0.2307692308 = 0.0000000000 0.0000000000 + z = -1.656973787 + -0.2307692308 z = -1.656973787 + -0.2307692308 Combine like terms: -1.656973787 + -0.2307692308 = -1.8877430178 z = -1.8877430178 Simplifying z = -1.8877430178Solution
The solution to the problem is based on the solutions from the subproblems. z = {1.4262045562, -1.8877430178}
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