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Simplifying 4n2 + 6n + -16 = -15n2 Reorder the terms: -16 + 6n + 4n2 = -15n2 Solving -16 + 6n + 4n2 = -15n2 Solving for variable 'n'. Combine like terms: 4n2 + 15n2 = 19n2 -16 + 6n + 19n2 = -15n2 + 15n2 Combine like terms: -15n2 + 15n2 = 0 -16 + 6n + 19n2 = 0 Begin completing the square. Divide all terms by 19 the coefficient of the squared term: Divide each side by '19'. -0.8421052632 + 0.3157894737n + n2 = 0 Move the constant term to the right: Add '0.8421052632' to each side of the equation. -0.8421052632 + 0.3157894737n + 0.8421052632 + n2 = 0 + 0.8421052632 Reorder the terms: -0.8421052632 + 0.8421052632 + 0.3157894737n + n2 = 0 + 0.8421052632 Combine like terms: -0.8421052632 + 0.8421052632 = 0.0000000000 0.0000000000 + 0.3157894737n + n2 = 0 + 0.8421052632 0.3157894737n + n2 = 0 + 0.8421052632 Combine like terms: 0 + 0.8421052632 = 0.8421052632 0.3157894737n + n2 = 0.8421052632 The n term is 0.3157894737n. Take half its coefficient (0.1578947369). Square it (0.02493074794) and add it to both sides. Add '0.02493074794' to each side of the equation. 0.3157894737n + 0.02493074794 + n2 = 0.8421052632 + 0.02493074794 Reorder the terms: 0.02493074794 + 0.3157894737n + n2 = 0.8421052632 + 0.02493074794 Combine like terms: 0.8421052632 + 0.02493074794 = 0.86703601114 0.02493074794 + 0.3157894737n + n2 = 0.86703601114 Factor a perfect square on the left side: (n + 0.1578947369)(n + 0.1578947369) = 0.86703601114 Calculate the square root of the right side: 0.931147685 Break this problem into two subproblems by setting (n + 0.1578947369) equal to 0.931147685 and -0.931147685.Subproblem 1
n + 0.1578947369 = 0.931147685 Simplifying n + 0.1578947369 = 0.931147685 Reorder the terms: 0.1578947369 + n = 0.931147685 Solving 0.1578947369 + n = 0.931147685 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.1578947369' to each side of the equation. 0.1578947369 + -0.1578947369 + n = 0.931147685 + -0.1578947369 Combine like terms: 0.1578947369 + -0.1578947369 = 0.0000000000 0.0000000000 + n = 0.931147685 + -0.1578947369 n = 0.931147685 + -0.1578947369 Combine like terms: 0.931147685 + -0.1578947369 = 0.7732529481 n = 0.7732529481 Simplifying n = 0.7732529481Subproblem 2
n + 0.1578947369 = -0.931147685 Simplifying n + 0.1578947369 = -0.931147685 Reorder the terms: 0.1578947369 + n = -0.931147685 Solving 0.1578947369 + n = -0.931147685 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.1578947369' to each side of the equation. 0.1578947369 + -0.1578947369 + n = -0.931147685 + -0.1578947369 Combine like terms: 0.1578947369 + -0.1578947369 = 0.0000000000 0.0000000000 + n = -0.931147685 + -0.1578947369 n = -0.931147685 + -0.1578947369 Combine like terms: -0.931147685 + -0.1578947369 = -1.0890424219 n = -1.0890424219 Simplifying n = -1.0890424219Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.7732529481, -1.0890424219}
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