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Simplifying (3 + -7n2) + (6n + 3) = 0 Remove parenthesis around (3 + -7n2) 3 + -7n2 + (6n + 3) = 0 Reorder the terms: 3 + -7n2 + (3 + 6n) = 0 Remove parenthesis around (3 + 6n) 3 + -7n2 + 3 + 6n = 0 Reorder the terms: 3 + 3 + 6n + -7n2 = 0 Combine like terms: 3 + 3 = 6 6 + 6n + -7n2 = 0 Solving 6 + 6n + -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 + -0.8571428571n + n2 = 0 Move the constant term to the right: Add '0.8571428571' to each side of the equation. -0.8571428571 + -0.8571428571n + 0.8571428571 + n2 = 0 + 0.8571428571 Reorder the terms: -0.8571428571 + 0.8571428571 + -0.8571428571n + n2 = 0 + 0.8571428571 Combine like terms: -0.8571428571 + 0.8571428571 = 0.0000000000 0.0000000000 + -0.8571428571n + n2 = 0 + 0.8571428571 -0.8571428571n + n2 = 0 + 0.8571428571 Combine like terms: 0 + 0.8571428571 = 0.8571428571 -0.8571428571n + n2 = 0.8571428571 The n term is -0.8571428571n. Take half its coefficient (-0.4285714286). Square it (0.1836734694) and add it to both sides. Add '0.1836734694' to each side of the equation. -0.8571428571n + 0.1836734694 + n2 = 0.8571428571 + 0.1836734694 Reorder the terms: 0.1836734694 + -0.8571428571n + n2 = 0.8571428571 + 0.1836734694 Combine like terms: 0.8571428571 + 0.1836734694 = 1.0408163265 0.1836734694 + -0.8571428571n + n2 = 1.0408163265 Factor a perfect square on the left side: (n + -0.4285714286)(n + -0.4285714286) = 1.0408163265 Calculate the square root of the right side: 1.020204061 Break this problem into two subproblems by setting (n + -0.4285714286) equal to 1.020204061 and -1.020204061.Subproblem 1
n + -0.4285714286 = 1.020204061 Simplifying n + -0.4285714286 = 1.020204061 Reorder the terms: -0.4285714286 + n = 1.020204061 Solving -0.4285714286 + n = 1.020204061 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.4285714286' to each side of the equation. -0.4285714286 + 0.4285714286 + n = 1.020204061 + 0.4285714286 Combine like terms: -0.4285714286 + 0.4285714286 = 0.0000000000 0.0000000000 + n = 1.020204061 + 0.4285714286 n = 1.020204061 + 0.4285714286 Combine like terms: 1.020204061 + 0.4285714286 = 1.4487754896 n = 1.4487754896 Simplifying n = 1.4487754896Subproblem 2
n + -0.4285714286 = -1.020204061 Simplifying n + -0.4285714286 = -1.020204061 Reorder the terms: -0.4285714286 + n = -1.020204061 Solving -0.4285714286 + n = -1.020204061 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.4285714286' to each side of the equation. -0.4285714286 + 0.4285714286 + n = -1.020204061 + 0.4285714286 Combine like terms: -0.4285714286 + 0.4285714286 = 0.0000000000 0.0000000000 + n = -1.020204061 + 0.4285714286 n = -1.020204061 + 0.4285714286 Combine like terms: -1.020204061 + 0.4285714286 = -0.5916326324 n = -0.5916326324 Simplifying n = -0.5916326324Solution
The solution to the problem is based on the solutions from the subproblems. n = {1.4487754896, -0.5916326324}
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