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Simplifying 5n2 + -13n + 7 = 0 Reorder the terms: 7 + -13n + 5n2 = 0 Solving 7 + -13n + 5n2 = 0 Solving for variable 'n'. Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. 1.4 + -2.6n + n2 = 0 Move the constant term to the right: Add '-1.4' to each side of the equation. 1.4 + -2.6n + -1.4 + n2 = 0 + -1.4 Reorder the terms: 1.4 + -1.4 + -2.6n + n2 = 0 + -1.4 Combine like terms: 1.4 + -1.4 = 0.0 0.0 + -2.6n + n2 = 0 + -1.4 -2.6n + n2 = 0 + -1.4 Combine like terms: 0 + -1.4 = -1.4 -2.6n + n2 = -1.4 The n term is -2.6n. Take half its coefficient (-1.3). Square it (1.69) and add it to both sides. Add '1.69' to each side of the equation. -2.6n + 1.69 + n2 = -1.4 + 1.69 Reorder the terms: 1.69 + -2.6n + n2 = -1.4 + 1.69 Combine like terms: -1.4 + 1.69 = 0.29 1.69 + -2.6n + n2 = 0.29 Factor a perfect square on the left side: (n + -1.3)(n + -1.3) = 0.29 Calculate the square root of the right side: 0.538516481 Break this problem into two subproblems by setting (n + -1.3) equal to 0.538516481 and -0.538516481.Subproblem 1
n + -1.3 = 0.538516481 Simplifying n + -1.3 = 0.538516481 Reorder the terms: -1.3 + n = 0.538516481 Solving -1.3 + n = 0.538516481 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.3' to each side of the equation. -1.3 + 1.3 + n = 0.538516481 + 1.3 Combine like terms: -1.3 + 1.3 = 0.0 0.0 + n = 0.538516481 + 1.3 n = 0.538516481 + 1.3 Combine like terms: 0.538516481 + 1.3 = 1.838516481 n = 1.838516481 Simplifying n = 1.838516481Subproblem 2
n + -1.3 = -0.538516481 Simplifying n + -1.3 = -0.538516481 Reorder the terms: -1.3 + n = -0.538516481 Solving -1.3 + n = -0.538516481 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.3' to each side of the equation. -1.3 + 1.3 + n = -0.538516481 + 1.3 Combine like terms: -1.3 + 1.3 = 0.0 0.0 + n = -0.538516481 + 1.3 n = -0.538516481 + 1.3 Combine like terms: -0.538516481 + 1.3 = 0.761483519 n = 0.761483519 Simplifying n = 0.761483519Solution
The solution to the problem is based on the solutions from the subproblems. n = {1.838516481, 0.761483519}
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