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Simplifying 12n2 + 15n + -18 = 1 Reorder the terms: -18 + 15n + 12n2 = 1 Solving -18 + 15n + 12n2 = 1 Solving for variable 'n'. Reorder the terms: -18 + -1 + 15n + 12n2 = 1 + -1 Combine like terms: -18 + -1 = -19 -19 + 15n + 12n2 = 1 + -1 Combine like terms: 1 + -1 = 0 -19 + 15n + 12n2 = 0 Begin completing the square. Divide all terms by 12 the coefficient of the squared term: Divide each side by '12'. -1.583333333 + 1.25n + n2 = 0 Move the constant term to the right: Add '1.583333333' to each side of the equation. -1.583333333 + 1.25n + 1.583333333 + n2 = 0 + 1.583333333 Reorder the terms: -1.583333333 + 1.583333333 + 1.25n + n2 = 0 + 1.583333333 Combine like terms: -1.583333333 + 1.583333333 = 0.000000000 0.000000000 + 1.25n + n2 = 0 + 1.583333333 1.25n + n2 = 0 + 1.583333333 Combine like terms: 0 + 1.583333333 = 1.583333333 1.25n + n2 = 1.583333333 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 = 1.583333333 + 0.390625 Reorder the terms: 0.390625 + 1.25n + n2 = 1.583333333 + 0.390625 Combine like terms: 1.583333333 + 0.390625 = 1.973958333 0.390625 + 1.25n + n2 = 1.973958333 Factor a perfect square on the left side: (n + 0.625)(n + 0.625) = 1.973958333 Calculate the square root of the right side: 1.404976275 Break this problem into two subproblems by setting (n + 0.625) equal to 1.404976275 and -1.404976275.Subproblem 1
n + 0.625 = 1.404976275 Simplifying n + 0.625 = 1.404976275 Reorder the terms: 0.625 + n = 1.404976275 Solving 0.625 + n = 1.404976275 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 = 1.404976275 + -0.625 Combine like terms: 0.625 + -0.625 = 0.000 0.000 + n = 1.404976275 + -0.625 n = 1.404976275 + -0.625 Combine like terms: 1.404976275 + -0.625 = 0.779976275 n = 0.779976275 Simplifying n = 0.779976275Subproblem 2
n + 0.625 = -1.404976275 Simplifying n + 0.625 = -1.404976275 Reorder the terms: 0.625 + n = -1.404976275 Solving 0.625 + n = -1.404976275 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 = -1.404976275 + -0.625 Combine like terms: 0.625 + -0.625 = 0.000 0.000 + n = -1.404976275 + -0.625 n = -1.404976275 + -0.625 Combine like terms: -1.404976275 + -0.625 = -2.029976275 n = -2.029976275 Simplifying n = -2.029976275Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.779976275, -2.029976275}
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