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Simplifying 6n2 + 3n + -27 = -11 Reorder the terms: -27 + 3n + 6n2 = -11 Solving -27 + 3n + 6n2 = -11 Solving for variable 'n'. Reorder the terms: -27 + 11 + 3n + 6n2 = -11 + 11 Combine like terms: -27 + 11 = -16 -16 + 3n + 6n2 = -11 + 11 Combine like terms: -11 + 11 = 0 -16 + 3n + 6n2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -2.666666667 + 0.5n + n2 = 0 Move the constant term to the right: Add '2.666666667' to each side of the equation. -2.666666667 + 0.5n + 2.666666667 + n2 = 0 + 2.666666667 Reorder the terms: -2.666666667 + 2.666666667 + 0.5n + n2 = 0 + 2.666666667 Combine like terms: -2.666666667 + 2.666666667 = 0.000000000 0.000000000 + 0.5n + n2 = 0 + 2.666666667 0.5n + n2 = 0 + 2.666666667 Combine like terms: 0 + 2.666666667 = 2.666666667 0.5n + n2 = 2.666666667 The n term is 0.5n. Take half its coefficient (0.25). Square it (0.0625) and add it to both sides. Add '0.0625' to each side of the equation. 0.5n + 0.0625 + n2 = 2.666666667 + 0.0625 Reorder the terms: 0.0625 + 0.5n + n2 = 2.666666667 + 0.0625 Combine like terms: 2.666666667 + 0.0625 = 2.729166667 0.0625 + 0.5n + n2 = 2.729166667 Factor a perfect square on the left side: (n + 0.25)(n + 0.25) = 2.729166667 Calculate the square root of the right side: 1.652018967 Break this problem into two subproblems by setting (n + 0.25) equal to 1.652018967 and -1.652018967.Subproblem 1
n + 0.25 = 1.652018967 Simplifying n + 0.25 = 1.652018967 Reorder the terms: 0.25 + n = 1.652018967 Solving 0.25 + n = 1.652018967 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + n = 1.652018967 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + n = 1.652018967 + -0.25 n = 1.652018967 + -0.25 Combine like terms: 1.652018967 + -0.25 = 1.402018967 n = 1.402018967 Simplifying n = 1.402018967Subproblem 2
n + 0.25 = -1.652018967 Simplifying n + 0.25 = -1.652018967 Reorder the terms: 0.25 + n = -1.652018967 Solving 0.25 + n = -1.652018967 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + n = -1.652018967 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + n = -1.652018967 + -0.25 n = -1.652018967 + -0.25 Combine like terms: -1.652018967 + -0.25 = -1.902018967 n = -1.902018967 Simplifying n = -1.902018967Solution
The solution to the problem is based on the solutions from the subproblems. n = {1.402018967, -1.902018967}
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