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Simplifying 1.5n + 0.8n(30 + -1n) = 54 1.5n + (30 * 0.8n + -1n * 0.8n) = 54 1.5n + (24n + -0.8n2) = 54 Combine like terms: 1.5n + 24n = 25.5n 25.5n + -0.8n2 = 54 Solving 25.5n + -0.8n2 = 54 Solving for variable 'n'. Reorder the terms: -54 + 25.5n + -0.8n2 = 54 + -54 Combine like terms: 54 + -54 = 0 -54 + 25.5n + -0.8n2 = 0 Begin completing the square. Divide all terms by -0.8 the coefficient of the squared term: Divide each side by '-0.8'. 67.5 + -31.875n + n2 = 0 Move the constant term to the right: Add '-67.5' to each side of the equation. 67.5 + -31.875n + -67.5 + n2 = 0 + -67.5 Reorder the terms: 67.5 + -67.5 + -31.875n + n2 = 0 + -67.5 Combine like terms: 67.5 + -67.5 = 0.0 0.0 + -31.875n + n2 = 0 + -67.5 -31.875n + n2 = 0 + -67.5 Combine like terms: 0 + -67.5 = -67.5 -31.875n + n2 = -67.5 The n term is -31.875n. Take half its coefficient (-15.9375). Square it (254.0039063) and add it to both sides. Add '254.0039063' to each side of the equation. -31.875n + 254.0039063 + n2 = -67.5 + 254.0039063 Reorder the terms: 254.0039063 + -31.875n + n2 = -67.5 + 254.0039063 Combine like terms: -67.5 + 254.0039063 = 186.5039063 254.0039063 + -31.875n + n2 = 186.5039063 Factor a perfect square on the left side: (n + -15.9375)(n + -15.9375) = 186.5039063 Calculate the square root of the right side: 13.656643303 Break this problem into two subproblems by setting (n + -15.9375) equal to 13.656643303 and -13.656643303.Subproblem 1
n + -15.9375 = 13.656643303 Simplifying n + -15.9375 = 13.656643303 Reorder the terms: -15.9375 + n = 13.656643303 Solving -15.9375 + n = 13.656643303 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '15.9375' to each side of the equation. -15.9375 + 15.9375 + n = 13.656643303 + 15.9375 Combine like terms: -15.9375 + 15.9375 = 0.0000 0.0000 + n = 13.656643303 + 15.9375 n = 13.656643303 + 15.9375 Combine like terms: 13.656643303 + 15.9375 = 29.594143303 n = 29.594143303 Simplifying n = 29.594143303Subproblem 2
n + -15.9375 = -13.656643303 Simplifying n + -15.9375 = -13.656643303 Reorder the terms: -15.9375 + n = -13.656643303 Solving -15.9375 + n = -13.656643303 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '15.9375' to each side of the equation. -15.9375 + 15.9375 + n = -13.656643303 + 15.9375 Combine like terms: -15.9375 + 15.9375 = 0.0000 0.0000 + n = -13.656643303 + 15.9375 n = -13.656643303 + 15.9375 Combine like terms: -13.656643303 + 15.9375 = 2.280856697 n = 2.280856697 Simplifying n = 2.280856697Solution
The solution to the problem is based on the solutions from the subproblems. n = {29.594143303, 2.280856697}
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