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Simplifying 85n2 + -56n + -144 = 0 Reorder the terms: -144 + -56n + 85n2 = 0 Solving -144 + -56n + 85n2 = 0 Solving for variable 'n'. Begin completing the square. Divide all terms by 85 the coefficient of the squared term: Divide each side by '85'. -1.694117647 + -0.6588235294n + n2 = 0 Move the constant term to the right: Add '1.694117647' to each side of the equation. -1.694117647 + -0.6588235294n + 1.694117647 + n2 = 0 + 1.694117647 Reorder the terms: -1.694117647 + 1.694117647 + -0.6588235294n + n2 = 0 + 1.694117647 Combine like terms: -1.694117647 + 1.694117647 = 0.000000000 0.000000000 + -0.6588235294n + n2 = 0 + 1.694117647 -0.6588235294n + n2 = 0 + 1.694117647 Combine like terms: 0 + 1.694117647 = 1.694117647 -0.6588235294n + n2 = 1.694117647 The n term is -0.6588235294n. Take half its coefficient (-0.3294117647). Square it (0.1085121107) and add it to both sides. Add '0.1085121107' to each side of the equation. -0.6588235294n + 0.1085121107 + n2 = 1.694117647 + 0.1085121107 Reorder the terms: 0.1085121107 + -0.6588235294n + n2 = 1.694117647 + 0.1085121107 Combine like terms: 1.694117647 + 0.1085121107 = 1.8026297577 0.1085121107 + -0.6588235294n + n2 = 1.8026297577 Factor a perfect square on the left side: (n + -0.3294117647)(n + -0.3294117647) = 1.8026297577 Calculate the square root of the right side: 1.342620482 Break this problem into two subproblems by setting (n + -0.3294117647) equal to 1.342620482 and -1.342620482.Subproblem 1
n + -0.3294117647 = 1.342620482 Simplifying n + -0.3294117647 = 1.342620482 Reorder the terms: -0.3294117647 + n = 1.342620482 Solving -0.3294117647 + n = 1.342620482 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.3294117647' to each side of the equation. -0.3294117647 + 0.3294117647 + n = 1.342620482 + 0.3294117647 Combine like terms: -0.3294117647 + 0.3294117647 = 0.0000000000 0.0000000000 + n = 1.342620482 + 0.3294117647 n = 1.342620482 + 0.3294117647 Combine like terms: 1.342620482 + 0.3294117647 = 1.6720322467 n = 1.6720322467 Simplifying n = 1.6720322467Subproblem 2
n + -0.3294117647 = -1.342620482 Simplifying n + -0.3294117647 = -1.342620482 Reorder the terms: -0.3294117647 + n = -1.342620482 Solving -0.3294117647 + n = -1.342620482 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.3294117647' to each side of the equation. -0.3294117647 + 0.3294117647 + n = -1.342620482 + 0.3294117647 Combine like terms: -0.3294117647 + 0.3294117647 = 0.0000000000 0.0000000000 + n = -1.342620482 + 0.3294117647 n = -1.342620482 + 0.3294117647 Combine like terms: -1.342620482 + 0.3294117647 = -1.0132087173 n = -1.0132087173 Simplifying n = -1.0132087173Solution
The solution to the problem is based on the solutions from the subproblems. n = {1.6720322467, -1.0132087173}
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