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Simplifying (-7n2 + 8n + -4) + -1(11n + 2 + -14n2) = 0 Reorder the terms: (-4 + 8n + -7n2) + -1(11n + 2 + -14n2) = 0 Remove parenthesis around (-4 + 8n + -7n2) -4 + 8n + -7n2 + -1(11n + 2 + -14n2) = 0 Reorder the terms: -4 + 8n + -7n2 + -1(2 + 11n + -14n2) = 0 -4 + 8n + -7n2 + (2 * -1 + 11n * -1 + -14n2 * -1) = 0 -4 + 8n + -7n2 + (-2 + -11n + 14n2) = 0 Reorder the terms: -4 + -2 + 8n + -11n + -7n2 + 14n2 = 0 Combine like terms: -4 + -2 = -6 -6 + 8n + -11n + -7n2 + 14n2 = 0 Combine like terms: 8n + -11n = -3n -6 + -3n + -7n2 + 14n2 = 0 Combine like terms: -7n2 + 14n2 = 7n2 -6 + -3n + 7n2 = 0 Solving -6 + -3n + 7n2 = 0 Solving for variable 'n'. Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. -0.8571428571 + -0.4285714286n + n2 = 0 Move the constant term to the right: Add '0.8571428571' to each side of the equation. -0.8571428571 + -0.4285714286n + 0.8571428571 + n2 = 0 + 0.8571428571 Reorder the terms: -0.8571428571 + 0.8571428571 + -0.4285714286n + n2 = 0 + 0.8571428571 Combine like terms: -0.8571428571 + 0.8571428571 = 0.0000000000 0.0000000000 + -0.4285714286n + n2 = 0 + 0.8571428571 -0.4285714286n + n2 = 0 + 0.8571428571 Combine like terms: 0 + 0.8571428571 = 0.8571428571 -0.4285714286n + n2 = 0.8571428571 The n term is -0.4285714286n. Take half its coefficient (-0.2142857143). Square it (0.04591836735) and add it to both sides. Add '0.04591836735' to each side of the equation. -0.4285714286n + 0.04591836735 + n2 = 0.8571428571 + 0.04591836735 Reorder the terms: 0.04591836735 + -0.4285714286n + n2 = 0.8571428571 + 0.04591836735 Combine like terms: 0.8571428571 + 0.04591836735 = 0.90306122445 0.04591836735 + -0.4285714286n + n2 = 0.90306122445 Factor a perfect square on the left side: (n + -0.2142857143)(n + -0.2142857143) = 0.90306122445 Calculate the square root of the right side: 0.950295335 Break this problem into two subproblems by setting (n + -0.2142857143) equal to 0.950295335 and -0.950295335.Subproblem 1
n + -0.2142857143 = 0.950295335 Simplifying n + -0.2142857143 = 0.950295335 Reorder the terms: -0.2142857143 + n = 0.950295335 Solving -0.2142857143 + n = 0.950295335 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.2142857143' to each side of the equation. -0.2142857143 + 0.2142857143 + n = 0.950295335 + 0.2142857143 Combine like terms: -0.2142857143 + 0.2142857143 = 0.0000000000 0.0000000000 + n = 0.950295335 + 0.2142857143 n = 0.950295335 + 0.2142857143 Combine like terms: 0.950295335 + 0.2142857143 = 1.1645810493 n = 1.1645810493 Simplifying n = 1.1645810493Subproblem 2
n + -0.2142857143 = -0.950295335 Simplifying n + -0.2142857143 = -0.950295335 Reorder the terms: -0.2142857143 + n = -0.950295335 Solving -0.2142857143 + n = -0.950295335 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.2142857143' to each side of the equation. -0.2142857143 + 0.2142857143 + n = -0.950295335 + 0.2142857143 Combine like terms: -0.2142857143 + 0.2142857143 = 0.0000000000 0.0000000000 + n = -0.950295335 + 0.2142857143 n = -0.950295335 + 0.2142857143 Combine like terms: -0.950295335 + 0.2142857143 = -0.7360096207 n = -0.7360096207 Simplifying n = -0.7360096207Solution
The solution to the problem is based on the solutions from the subproblems. n = {1.1645810493, -0.7360096207}
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