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Simplifying 8n2 + 12n + -7 = 10 Reorder the terms: -7 + 12n + 8n2 = 10 Solving -7 + 12n + 8n2 = 10 Solving for variable 'n'. Reorder the terms: -7 + -10 + 12n + 8n2 = 10 + -10 Combine like terms: -7 + -10 = -17 -17 + 12n + 8n2 = 10 + -10 Combine like terms: 10 + -10 = 0 -17 + 12n + 8n2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. -2.125 + 1.5n + n2 = 0 Move the constant term to the right: Add '2.125' to each side of the equation. -2.125 + 1.5n + 2.125 + n2 = 0 + 2.125 Reorder the terms: -2.125 + 2.125 + 1.5n + n2 = 0 + 2.125 Combine like terms: -2.125 + 2.125 = 0.000 0.000 + 1.5n + n2 = 0 + 2.125 1.5n + n2 = 0 + 2.125 Combine like terms: 0 + 2.125 = 2.125 1.5n + n2 = 2.125 The n term is 1.5n. Take half its coefficient (0.75). Square it (0.5625) and add it to both sides. Add '0.5625' to each side of the equation. 1.5n + 0.5625 + n2 = 2.125 + 0.5625 Reorder the terms: 0.5625 + 1.5n + n2 = 2.125 + 0.5625 Combine like terms: 2.125 + 0.5625 = 2.6875 0.5625 + 1.5n + n2 = 2.6875 Factor a perfect square on the left side: (n + 0.75)(n + 0.75) = 2.6875 Calculate the square root of the right side: 1.639359631 Break this problem into two subproblems by setting (n + 0.75) equal to 1.639359631 and -1.639359631.Subproblem 1
n + 0.75 = 1.639359631 Simplifying n + 0.75 = 1.639359631 Reorder the terms: 0.75 + n = 1.639359631 Solving 0.75 + n = 1.639359631 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + n = 1.639359631 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + n = 1.639359631 + -0.75 n = 1.639359631 + -0.75 Combine like terms: 1.639359631 + -0.75 = 0.889359631 n = 0.889359631 Simplifying n = 0.889359631Subproblem 2
n + 0.75 = -1.639359631 Simplifying n + 0.75 = -1.639359631 Reorder the terms: 0.75 + n = -1.639359631 Solving 0.75 + n = -1.639359631 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + n = -1.639359631 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + n = -1.639359631 + -0.75 n = -1.639359631 + -0.75 Combine like terms: -1.639359631 + -0.75 = -2.389359631 n = -2.389359631 Simplifying n = -2.389359631Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.889359631, -2.389359631}
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