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Simplifying 5n2 = 3n + 10 Reorder the terms: 5n2 = 10 + 3n Solving 5n2 = 10 + 3n Solving for variable 'n'. Reorder the terms: -10 + -3n + 5n2 = 10 + 3n + -10 + -3n Reorder the terms: -10 + -3n + 5n2 = 10 + -10 + 3n + -3n Combine like terms: 10 + -10 = 0 -10 + -3n + 5n2 = 0 + 3n + -3n -10 + -3n + 5n2 = 3n + -3n Combine like terms: 3n + -3n = 0 -10 + -3n + 5n2 = 0 Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. -2 + -0.6n + n2 = 0 Move the constant term to the right: Add '2' to each side of the equation. -2 + -0.6n + 2 + n2 = 0 + 2 Reorder the terms: -2 + 2 + -0.6n + n2 = 0 + 2 Combine like terms: -2 + 2 = 0 0 + -0.6n + n2 = 0 + 2 -0.6n + n2 = 0 + 2 Combine like terms: 0 + 2 = 2 -0.6n + n2 = 2 The n term is -0.6n. Take half its coefficient (-0.3). Square it (0.09) and add it to both sides. Add '0.09' to each side of the equation. -0.6n + 0.09 + n2 = 2 + 0.09 Reorder the terms: 0.09 + -0.6n + n2 = 2 + 0.09 Combine like terms: 2 + 0.09 = 2.09 0.09 + -0.6n + n2 = 2.09 Factor a perfect square on the left side: (n + -0.3)(n + -0.3) = 2.09 Calculate the square root of the right side: 1.445683229 Break this problem into two subproblems by setting (n + -0.3) equal to 1.445683229 and -1.445683229.Subproblem 1
n + -0.3 = 1.445683229 Simplifying n + -0.3 = 1.445683229 Reorder the terms: -0.3 + n = 1.445683229 Solving -0.3 + n = 1.445683229 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.3' to each side of the equation. -0.3 + 0.3 + n = 1.445683229 + 0.3 Combine like terms: -0.3 + 0.3 = 0.0 0.0 + n = 1.445683229 + 0.3 n = 1.445683229 + 0.3 Combine like terms: 1.445683229 + 0.3 = 1.745683229 n = 1.745683229 Simplifying n = 1.745683229Subproblem 2
n + -0.3 = -1.445683229 Simplifying n + -0.3 = -1.445683229 Reorder the terms: -0.3 + n = -1.445683229 Solving -0.3 + n = -1.445683229 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.3' to each side of the equation. -0.3 + 0.3 + n = -1.445683229 + 0.3 Combine like terms: -0.3 + 0.3 = 0.0 0.0 + n = -1.445683229 + 0.3 n = -1.445683229 + 0.3 Combine like terms: -1.445683229 + 0.3 = -1.145683229 n = -1.145683229 Simplifying n = -1.145683229Solution
The solution to the problem is based on the solutions from the subproblems. n = {1.745683229, -1.145683229}
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