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Simplifying 9n2 + -6n = 17 Reorder the terms: -6n + 9n2 = 17 Solving -6n + 9n2 = 17 Solving for variable 'n'. Reorder the terms: -17 + -6n + 9n2 = 17 + -17 Combine like terms: 17 + -17 = 0 -17 + -6n + 9n2 = 0 Begin completing the square. Divide all terms by 9 the coefficient of the squared term: Divide each side by '9'. -1.888888889 + -0.6666666667n + n2 = 0 Move the constant term to the right: Add '1.888888889' to each side of the equation. -1.888888889 + -0.6666666667n + 1.888888889 + n2 = 0 + 1.888888889 Reorder the terms: -1.888888889 + 1.888888889 + -0.6666666667n + n2 = 0 + 1.888888889 Combine like terms: -1.888888889 + 1.888888889 = 0.000000000 0.000000000 + -0.6666666667n + n2 = 0 + 1.888888889 -0.6666666667n + n2 = 0 + 1.888888889 Combine like terms: 0 + 1.888888889 = 1.888888889 -0.6666666667n + n2 = 1.888888889 The n term is -0.6666666667n. Take half its coefficient (-0.3333333334). Square it (0.1111111112) and add it to both sides. Add '0.1111111112' to each side of the equation. -0.6666666667n + 0.1111111112 + n2 = 1.888888889 + 0.1111111112 Reorder the terms: 0.1111111112 + -0.6666666667n + n2 = 1.888888889 + 0.1111111112 Combine like terms: 1.888888889 + 0.1111111112 = 2.0000000002 0.1111111112 + -0.6666666667n + n2 = 2.0000000002 Factor a perfect square on the left side: (n + -0.3333333334)(n + -0.3333333334) = 2.0000000002 Calculate the square root of the right side: 1.414213562 Break this problem into two subproblems by setting (n + -0.3333333334) equal to 1.414213562 and -1.414213562.Subproblem 1
n + -0.3333333334 = 1.414213562 Simplifying n + -0.3333333334 = 1.414213562 Reorder the terms: -0.3333333334 + n = 1.414213562 Solving -0.3333333334 + n = 1.414213562 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.3333333334' to each side of the equation. -0.3333333334 + 0.3333333334 + n = 1.414213562 + 0.3333333334 Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000 0.0000000000 + n = 1.414213562 + 0.3333333334 n = 1.414213562 + 0.3333333334 Combine like terms: 1.414213562 + 0.3333333334 = 1.7475468954 n = 1.7475468954 Simplifying n = 1.7475468954Subproblem 2
n + -0.3333333334 = -1.414213562 Simplifying n + -0.3333333334 = -1.414213562 Reorder the terms: -0.3333333334 + n = -1.414213562 Solving -0.3333333334 + n = -1.414213562 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.3333333334' to each side of the equation. -0.3333333334 + 0.3333333334 + n = -1.414213562 + 0.3333333334 Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000 0.0000000000 + n = -1.414213562 + 0.3333333334 n = -1.414213562 + 0.3333333334 Combine like terms: -1.414213562 + 0.3333333334 = -1.0808802286 n = -1.0808802286 Simplifying n = -1.0808802286Solution
The solution to the problem is based on the solutions from the subproblems. n = {1.7475468954, -1.0808802286}
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