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Simplifying 4n2 + -9n + -18 = 6 Reorder the terms: -18 + -9n + 4n2 = 6 Solving -18 + -9n + 4n2 = 6 Solving for variable 'n'. Reorder the terms: -18 + -6 + -9n + 4n2 = 6 + -6 Combine like terms: -18 + -6 = -24 -24 + -9n + 4n2 = 6 + -6 Combine like terms: 6 + -6 = 0 -24 + -9n + 4n2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -6 + -2.25n + n2 = 0 Move the constant term to the right: Add '6' to each side of the equation. -6 + -2.25n + 6 + n2 = 0 + 6 Reorder the terms: -6 + 6 + -2.25n + n2 = 0 + 6 Combine like terms: -6 + 6 = 0 0 + -2.25n + n2 = 0 + 6 -2.25n + n2 = 0 + 6 Combine like terms: 0 + 6 = 6 -2.25n + n2 = 6 The n term is -2.25n. Take half its coefficient (-1.125). Square it (1.265625) and add it to both sides. Add '1.265625' to each side of the equation. -2.25n + 1.265625 + n2 = 6 + 1.265625 Reorder the terms: 1.265625 + -2.25n + n2 = 6 + 1.265625 Combine like terms: 6 + 1.265625 = 7.265625 1.265625 + -2.25n + n2 = 7.265625 Factor a perfect square on the left side: (n + -1.125)(n + -1.125) = 7.265625 Calculate the square root of the right side: 2.695482332 Break this problem into two subproblems by setting (n + -1.125) equal to 2.695482332 and -2.695482332.Subproblem 1
n + -1.125 = 2.695482332 Simplifying n + -1.125 = 2.695482332 Reorder the terms: -1.125 + n = 2.695482332 Solving -1.125 + n = 2.695482332 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.125' to each side of the equation. -1.125 + 1.125 + n = 2.695482332 + 1.125 Combine like terms: -1.125 + 1.125 = 0.000 0.000 + n = 2.695482332 + 1.125 n = 2.695482332 + 1.125 Combine like terms: 2.695482332 + 1.125 = 3.820482332 n = 3.820482332 Simplifying n = 3.820482332Subproblem 2
n + -1.125 = -2.695482332 Simplifying n + -1.125 = -2.695482332 Reorder the terms: -1.125 + n = -2.695482332 Solving -1.125 + n = -2.695482332 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.125' to each side of the equation. -1.125 + 1.125 + n = -2.695482332 + 1.125 Combine like terms: -1.125 + 1.125 = 0.000 0.000 + n = -2.695482332 + 1.125 n = -2.695482332 + 1.125 Combine like terms: -2.695482332 + 1.125 = -1.570482332 n = -1.570482332 Simplifying n = -1.570482332Solution
The solution to the problem is based on the solutions from the subproblems. n = {3.820482332, -1.570482332}
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