28n^2=96-184n

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Solution for 28n^2=96-184n equation:


Simplifying
28n2 = 96 + -184n

Solving
28n2 = 96 + -184n

Solving for variable 'n'.

Reorder the terms:
-96 + 184n + 28n2 = 96 + -184n + -96 + 184n

Reorder the terms:
-96 + 184n + 28n2 = 96 + -96 + -184n + 184n

Combine like terms: 96 + -96 = 0
-96 + 184n + 28n2 = 0 + -184n + 184n
-96 + 184n + 28n2 = -184n + 184n

Combine like terms: -184n + 184n = 0
-96 + 184n + 28n2 = 0

Factor out the Greatest Common Factor (GCF), '4'.
4(-24 + 46n + 7n2) = 0

Ignore the factor 4.

Subproblem 1

Set the factor '(-24 + 46n + 7n2)' equal to zero and attempt to solve: Simplifying -24 + 46n + 7n2 = 0 Solving -24 + 46n + 7n2 = 0 Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. -3.428571429 + 6.571428571n + n2 = 0 Move the constant term to the right: Add '3.428571429' to each side of the equation. -3.428571429 + 6.571428571n + 3.428571429 + n2 = 0 + 3.428571429 Reorder the terms: -3.428571429 + 3.428571429 + 6.571428571n + n2 = 0 + 3.428571429 Combine like terms: -3.428571429 + 3.428571429 = 0.000000000 0.000000000 + 6.571428571n + n2 = 0 + 3.428571429 6.571428571n + n2 = 0 + 3.428571429 Combine like terms: 0 + 3.428571429 = 3.428571429 6.571428571n + n2 = 3.428571429 The n term is 6.571428571n. Take half its coefficient (3.285714286). Square it (10.79591837) and add it to both sides. Add '10.79591837' to each side of the equation. 6.571428571n + 10.79591837 + n2 = 3.428571429 + 10.79591837 Reorder the terms: 10.79591837 + 6.571428571n + n2 = 3.428571429 + 10.79591837 Combine like terms: 3.428571429 + 10.79591837 = 14.224489799 10.79591837 + 6.571428571n + n2 = 14.224489799 Factor a perfect square on the left side: (n + 3.285714286)(n + 3.285714286) = 14.224489799 Calculate the square root of the right side: 3.771536795 Break this problem into two subproblems by setting (n + 3.285714286) equal to 3.771536795 and -3.771536795.

Subproblem 1

n + 3.285714286 = 3.771536795 Simplifying n + 3.285714286 = 3.771536795 Reorder the terms: 3.285714286 + n = 3.771536795 Solving 3.285714286 + n = 3.771536795 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-3.285714286' to each side of the equation. 3.285714286 + -3.285714286 + n = 3.771536795 + -3.285714286 Combine like terms: 3.285714286 + -3.285714286 = 0.000000000 0.000000000 + n = 3.771536795 + -3.285714286 n = 3.771536795 + -3.285714286 Combine like terms: 3.771536795 + -3.285714286 = 0.485822509 n = 0.485822509 Simplifying n = 0.485822509

Subproblem 2

n + 3.285714286 = -3.771536795 Simplifying n + 3.285714286 = -3.771536795 Reorder the terms: 3.285714286 + n = -3.771536795 Solving 3.285714286 + n = -3.771536795 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-3.285714286' to each side of the equation. 3.285714286 + -3.285714286 + n = -3.771536795 + -3.285714286 Combine like terms: 3.285714286 + -3.285714286 = 0.000000000 0.000000000 + n = -3.771536795 + -3.285714286 n = -3.771536795 + -3.285714286 Combine like terms: -3.771536795 + -3.285714286 = -7.057251081 n = -7.057251081 Simplifying n = -7.057251081

Solution

The solution to the problem is based on the solutions from the subproblems. n = {0.485822509, -7.057251081}

Solution

n = {0.485822509, -7.057251081}

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