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Simplifying 7n2 = -21n + 16 Reorder the terms: 7n2 = 16 + -21n Solving 7n2 = 16 + -21n Solving for variable 'n'. Reorder the terms: -16 + 21n + 7n2 = 16 + -21n + -16 + 21n Reorder the terms: -16 + 21n + 7n2 = 16 + -16 + -21n + 21n Combine like terms: 16 + -16 = 0 -16 + 21n + 7n2 = 0 + -21n + 21n -16 + 21n + 7n2 = -21n + 21n Combine like terms: -21n + 21n = 0 -16 + 21n + 7n2 = 0 Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. -2.285714286 + 3n + n2 = 0 Move the constant term to the right: Add '2.285714286' to each side of the equation. -2.285714286 + 3n + 2.285714286 + n2 = 0 + 2.285714286 Reorder the terms: -2.285714286 + 2.285714286 + 3n + n2 = 0 + 2.285714286 Combine like terms: -2.285714286 + 2.285714286 = 0.000000000 0.000000000 + 3n + n2 = 0 + 2.285714286 3n + n2 = 0 + 2.285714286 Combine like terms: 0 + 2.285714286 = 2.285714286 3n + n2 = 2.285714286 The n term is 3n. Take half its coefficient (1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. 3n + 2.25 + n2 = 2.285714286 + 2.25 Reorder the terms: 2.25 + 3n + n2 = 2.285714286 + 2.25 Combine like terms: 2.285714286 + 2.25 = 4.535714286 2.25 + 3n + n2 = 4.535714286 Factor a perfect square on the left side: (n + 1.5)(n + 1.5) = 4.535714286 Calculate the square root of the right side: 2.129721645 Break this problem into two subproblems by setting (n + 1.5) equal to 2.129721645 and -2.129721645.Subproblem 1
n + 1.5 = 2.129721645 Simplifying n + 1.5 = 2.129721645 Reorder the terms: 1.5 + n = 2.129721645 Solving 1.5 + n = 2.129721645 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + n = 2.129721645 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + n = 2.129721645 + -1.5 n = 2.129721645 + -1.5 Combine like terms: 2.129721645 + -1.5 = 0.629721645 n = 0.629721645 Simplifying n = 0.629721645Subproblem 2
n + 1.5 = -2.129721645 Simplifying n + 1.5 = -2.129721645 Reorder the terms: 1.5 + n = -2.129721645 Solving 1.5 + n = -2.129721645 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + n = -2.129721645 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + n = -2.129721645 + -1.5 n = -2.129721645 + -1.5 Combine like terms: -2.129721645 + -1.5 = -3.629721645 n = -3.629721645 Simplifying n = -3.629721645Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.629721645, -3.629721645}
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