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Simplifying -8n2 + 7n + 8 = 9 Reorder the terms: 8 + 7n + -8n2 = 9 Solving 8 + 7n + -8n2 = 9 Solving for variable 'n'. Reorder the terms: 8 + -9 + 7n + -8n2 = 9 + -9 Combine like terms: 8 + -9 = -1 -1 + 7n + -8n2 = 9 + -9 Combine like terms: 9 + -9 = 0 -1 + 7n + -8n2 = 0 Begin completing the square. Divide all terms by -8 the coefficient of the squared term: Divide each side by '-8'. 0.125 + -0.875n + n2 = 0 Move the constant term to the right: Add '-0.125' to each side of the equation. 0.125 + -0.875n + -0.125 + n2 = 0 + -0.125 Reorder the terms: 0.125 + -0.125 + -0.875n + n2 = 0 + -0.125 Combine like terms: 0.125 + -0.125 = 0.000 0.000 + -0.875n + n2 = 0 + -0.125 -0.875n + n2 = 0 + -0.125 Combine like terms: 0 + -0.125 = -0.125 -0.875n + n2 = -0.125 The n term is -0.875n. Take half its coefficient (-0.4375). Square it (0.19140625) and add it to both sides. Add '0.19140625' to each side of the equation. -0.875n + 0.19140625 + n2 = -0.125 + 0.19140625 Reorder the terms: 0.19140625 + -0.875n + n2 = -0.125 + 0.19140625 Combine like terms: -0.125 + 0.19140625 = 0.06640625 0.19140625 + -0.875n + n2 = 0.06640625 Factor a perfect square on the left side: (n + -0.4375)(n + -0.4375) = 0.06640625 Calculate the square root of the right side: 0.257694102 Break this problem into two subproblems by setting (n + -0.4375) equal to 0.257694102 and -0.257694102.Subproblem 1
n + -0.4375 = 0.257694102 Simplifying n + -0.4375 = 0.257694102 Reorder the terms: -0.4375 + n = 0.257694102 Solving -0.4375 + n = 0.257694102 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.4375' to each side of the equation. -0.4375 + 0.4375 + n = 0.257694102 + 0.4375 Combine like terms: -0.4375 + 0.4375 = 0.0000 0.0000 + n = 0.257694102 + 0.4375 n = 0.257694102 + 0.4375 Combine like terms: 0.257694102 + 0.4375 = 0.695194102 n = 0.695194102 Simplifying n = 0.695194102Subproblem 2
n + -0.4375 = -0.257694102 Simplifying n + -0.4375 = -0.257694102 Reorder the terms: -0.4375 + n = -0.257694102 Solving -0.4375 + n = -0.257694102 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.4375' to each side of the equation. -0.4375 + 0.4375 + n = -0.257694102 + 0.4375 Combine like terms: -0.4375 + 0.4375 = 0.0000 0.0000 + n = -0.257694102 + 0.4375 n = -0.257694102 + 0.4375 Combine like terms: -0.257694102 + 0.4375 = 0.179805898 n = 0.179805898 Simplifying n = 0.179805898Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.695194102, 0.179805898}
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