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Simplifying 4n2 + 5n + -420 = 0 Reorder the terms: -420 + 5n + 4n2 = 0 Solving -420 + 5n + 4n2 = 0 Solving for variable 'n'. Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -105 + 1.25n + n2 = 0 Move the constant term to the right: Add '105' to each side of the equation. -105 + 1.25n + 105 + n2 = 0 + 105 Reorder the terms: -105 + 105 + 1.25n + n2 = 0 + 105 Combine like terms: -105 + 105 = 0 0 + 1.25n + n2 = 0 + 105 1.25n + n2 = 0 + 105 Combine like terms: 0 + 105 = 105 1.25n + n2 = 105 The n term is 1.25n. Take half its coefficient (0.625). Square it (0.390625) and add it to both sides. Add '0.390625' to each side of the equation. 1.25n + 0.390625 + n2 = 105 + 0.390625 Reorder the terms: 0.390625 + 1.25n + n2 = 105 + 0.390625 Combine like terms: 105 + 0.390625 = 105.390625 0.390625 + 1.25n + n2 = 105.390625 Factor a perfect square on the left side: (n + 0.625)(n + 0.625) = 105.390625 Calculate the square root of the right side: 10.26599362 Break this problem into two subproblems by setting (n + 0.625) equal to 10.26599362 and -10.26599362.Subproblem 1
n + 0.625 = 10.26599362 Simplifying n + 0.625 = 10.26599362 Reorder the terms: 0.625 + n = 10.26599362 Solving 0.625 + n = 10.26599362 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.625' to each side of the equation. 0.625 + -0.625 + n = 10.26599362 + -0.625 Combine like terms: 0.625 + -0.625 = 0.000 0.000 + n = 10.26599362 + -0.625 n = 10.26599362 + -0.625 Combine like terms: 10.26599362 + -0.625 = 9.64099362 n = 9.64099362 Simplifying n = 9.64099362Subproblem 2
n + 0.625 = -10.26599362 Simplifying n + 0.625 = -10.26599362 Reorder the terms: 0.625 + n = -10.26599362 Solving 0.625 + n = -10.26599362 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.625' to each side of the equation. 0.625 + -0.625 + n = -10.26599362 + -0.625 Combine like terms: 0.625 + -0.625 = 0.000 0.000 + n = -10.26599362 + -0.625 n = -10.26599362 + -0.625 Combine like terms: -10.26599362 + -0.625 = -10.89099362 n = -10.89099362 Simplifying n = -10.89099362Solution
The solution to the problem is based on the solutions from the subproblems. n = {9.64099362, -10.89099362}
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