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Simplifying 7n(-3 + 5n) + 2 = 0 (-3 * 7n + 5n * 7n) + 2 = 0 (-21n + 35n2) + 2 = 0 Reorder the terms: 2 + -21n + 35n2 = 0 Solving 2 + -21n + 35n2 = 0 Solving for variable 'n'. Begin completing the square. Divide all terms by 35 the coefficient of the squared term: Divide each side by '35'. 0.05714285714 + -0.6n + n2 = 0 Move the constant term to the right: Add '-0.05714285714' to each side of the equation. 0.05714285714 + -0.6n + -0.05714285714 + n2 = 0 + -0.05714285714 Reorder the terms: 0.05714285714 + -0.05714285714 + -0.6n + n2 = 0 + -0.05714285714 Combine like terms: 0.05714285714 + -0.05714285714 = 0.00000000000 0.00000000000 + -0.6n + n2 = 0 + -0.05714285714 -0.6n + n2 = 0 + -0.05714285714 Combine like terms: 0 + -0.05714285714 = -0.05714285714 -0.6n + n2 = -0.05714285714 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 = -0.05714285714 + 0.09 Reorder the terms: 0.09 + -0.6n + n2 = -0.05714285714 + 0.09 Combine like terms: -0.05714285714 + 0.09 = 0.03285714286 0.09 + -0.6n + n2 = 0.03285714286 Factor a perfect square on the left side: (n + -0.3)(n + -0.3) = 0.03285714286 Calculate the square root of the right side: 0.181265393 Break this problem into two subproblems by setting (n + -0.3) equal to 0.181265393 and -0.181265393.Subproblem 1
n + -0.3 = 0.181265393 Simplifying n + -0.3 = 0.181265393 Reorder the terms: -0.3 + n = 0.181265393 Solving -0.3 + n = 0.181265393 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 = 0.181265393 + 0.3 Combine like terms: -0.3 + 0.3 = 0.0 0.0 + n = 0.181265393 + 0.3 n = 0.181265393 + 0.3 Combine like terms: 0.181265393 + 0.3 = 0.481265393 n = 0.481265393 Simplifying n = 0.481265393Subproblem 2
n + -0.3 = -0.181265393 Simplifying n + -0.3 = -0.181265393 Reorder the terms: -0.3 + n = -0.181265393 Solving -0.3 + n = -0.181265393 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 = -0.181265393 + 0.3 Combine like terms: -0.3 + 0.3 = 0.0 0.0 + n = -0.181265393 + 0.3 n = -0.181265393 + 0.3 Combine like terms: -0.181265393 + 0.3 = 0.118734607 n = 0.118734607 Simplifying n = 0.118734607Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.481265393, 0.118734607}
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