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Simplifying 7n2 + 9n + -12 = 0 Reorder the terms: -12 + 9n + 7n2 = 0 Solving -12 + 9n + 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'. -1.714285714 + 1.285714286n + n2 = 0 Move the constant term to the right: Add '1.714285714' to each side of the equation. -1.714285714 + 1.285714286n + 1.714285714 + n2 = 0 + 1.714285714 Reorder the terms: -1.714285714 + 1.714285714 + 1.285714286n + n2 = 0 + 1.714285714 Combine like terms: -1.714285714 + 1.714285714 = 0.000000000 0.000000000 + 1.285714286n + n2 = 0 + 1.714285714 1.285714286n + n2 = 0 + 1.714285714 Combine like terms: 0 + 1.714285714 = 1.714285714 1.285714286n + n2 = 1.714285714 The n term is 1.285714286n. Take half its coefficient (0.642857143). Square it (0.4132653063) and add it to both sides. Add '0.4132653063' to each side of the equation. 1.285714286n + 0.4132653063 + n2 = 1.714285714 + 0.4132653063 Reorder the terms: 0.4132653063 + 1.285714286n + n2 = 1.714285714 + 0.4132653063 Combine like terms: 1.714285714 + 0.4132653063 = 2.1275510203 0.4132653063 + 1.285714286n + n2 = 2.1275510203 Factor a perfect square on the left side: (n + 0.642857143)(n + 0.642857143) = 2.1275510203 Calculate the square root of the right side: 1.458612704 Break this problem into two subproblems by setting (n + 0.642857143) equal to 1.458612704 and -1.458612704.Subproblem 1
n + 0.642857143 = 1.458612704 Simplifying n + 0.642857143 = 1.458612704 Reorder the terms: 0.642857143 + n = 1.458612704 Solving 0.642857143 + n = 1.458612704 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.642857143' to each side of the equation. 0.642857143 + -0.642857143 + n = 1.458612704 + -0.642857143 Combine like terms: 0.642857143 + -0.642857143 = 0.000000000 0.000000000 + n = 1.458612704 + -0.642857143 n = 1.458612704 + -0.642857143 Combine like terms: 1.458612704 + -0.642857143 = 0.815755561 n = 0.815755561 Simplifying n = 0.815755561Subproblem 2
n + 0.642857143 = -1.458612704 Simplifying n + 0.642857143 = -1.458612704 Reorder the terms: 0.642857143 + n = -1.458612704 Solving 0.642857143 + n = -1.458612704 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.642857143' to each side of the equation. 0.642857143 + -0.642857143 + n = -1.458612704 + -0.642857143 Combine like terms: 0.642857143 + -0.642857143 = 0.000000000 0.000000000 + n = -1.458612704 + -0.642857143 n = -1.458612704 + -0.642857143 Combine like terms: -1.458612704 + -0.642857143 = -2.101469847 n = -2.101469847 Simplifying n = -2.101469847Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.815755561, -2.101469847}
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