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Simplifying 5n2 + 12n + 6 = 0 Reorder the terms: 6 + 12n + 5n2 = 0 Solving 6 + 12n + 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'. 1.2 + 2.4n + n2 = 0 Move the constant term to the right: Add '-1.2' to each side of the equation. 1.2 + 2.4n + -1.2 + n2 = 0 + -1.2 Reorder the terms: 1.2 + -1.2 + 2.4n + n2 = 0 + -1.2 Combine like terms: 1.2 + -1.2 = 0.0 0.0 + 2.4n + n2 = 0 + -1.2 2.4n + n2 = 0 + -1.2 Combine like terms: 0 + -1.2 = -1.2 2.4n + n2 = -1.2 The n term is 2.4n. Take half its coefficient (1.2). Square it (1.44) and add it to both sides. Add '1.44' to each side of the equation. 2.4n + 1.44 + n2 = -1.2 + 1.44 Reorder the terms: 1.44 + 2.4n + n2 = -1.2 + 1.44 Combine like terms: -1.2 + 1.44 = 0.24 1.44 + 2.4n + n2 = 0.24 Factor a perfect square on the left side: (n + 1.2)(n + 1.2) = 0.24 Calculate the square root of the right side: 0.489897949 Break this problem into two subproblems by setting (n + 1.2) equal to 0.489897949 and -0.489897949.Subproblem 1
n + 1.2 = 0.489897949 Simplifying n + 1.2 = 0.489897949 Reorder the terms: 1.2 + n = 0.489897949 Solving 1.2 + n = 0.489897949 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.2' to each side of the equation. 1.2 + -1.2 + n = 0.489897949 + -1.2 Combine like terms: 1.2 + -1.2 = 0.0 0.0 + n = 0.489897949 + -1.2 n = 0.489897949 + -1.2 Combine like terms: 0.489897949 + -1.2 = -0.710102051 n = -0.710102051 Simplifying n = -0.710102051Subproblem 2
n + 1.2 = -0.489897949 Simplifying n + 1.2 = -0.489897949 Reorder the terms: 1.2 + n = -0.489897949 Solving 1.2 + n = -0.489897949 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.2' to each side of the equation. 1.2 + -1.2 + n = -0.489897949 + -1.2 Combine like terms: 1.2 + -1.2 = 0.0 0.0 + n = -0.489897949 + -1.2 n = -0.489897949 + -1.2 Combine like terms: -0.489897949 + -1.2 = -1.689897949 n = -1.689897949 Simplifying n = -1.689897949Solution
The solution to the problem is based on the solutions from the subproblems. n = {-0.710102051, -1.689897949}
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