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Simplifying 0.5(n + 2)(n + -3) = 12 Reorder the terms: 0.5(2 + n)(n + -3) = 12 Reorder the terms: 0.5(2 + n)(-3 + n) = 12 Multiply (2 + n) * (-3 + n) 0.5(2(-3 + n) + n(-3 + n)) = 12 0.5((-3 * 2 + n * 2) + n(-3 + n)) = 12 0.5((-6 + 2n) + n(-3 + n)) = 12 0.5(-6 + 2n + (-3 * n + n * n)) = 12 0.5(-6 + 2n + (-3n + n2)) = 12 Combine like terms: 2n + -3n = -1n 0.5(-6 + -1n + n2) = 12 (-6 * 0.5 + -1n * 0.5 + n2 * 0.5) = 12 (-3 + -0.5n + 0.5n2) = 12 Solving -3 + -0.5n + 0.5n2 = 12 Solving for variable 'n'. Reorder the terms: -3 + -12 + -0.5n + 0.5n2 = 12 + -12 Combine like terms: -3 + -12 = -15 -15 + -0.5n + 0.5n2 = 12 + -12 Combine like terms: 12 + -12 = 0 -15 + -0.5n + 0.5n2 = 0 Begin completing the square. Divide all terms by 0.5 the coefficient of the squared term: Divide each side by '0.5'. -30 + -1n + n2 = 0 Move the constant term to the right: Add '30' to each side of the equation. -30 + -1n + 30 + n2 = 0 + 30 Reorder the terms: -30 + 30 + -1n + n2 = 0 + 30 Combine like terms: -30 + 30 = 0 0 + -1n + n2 = 0 + 30 -1n + n2 = 0 + 30 Combine like terms: 0 + 30 = 30 -1n + n2 = 30 The n term is -1n. Take half its coefficient (-0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. -1n + 0.25 + n2 = 30 + 0.25 Reorder the terms: 0.25 + -1n + n2 = 30 + 0.25 Combine like terms: 30 + 0.25 = 30.25 0.25 + -1n + n2 = 30.25 Factor a perfect square on the left side: (n + -0.5)(n + -0.5) = 30.25 Calculate the square root of the right side: 5.5 Break this problem into two subproblems by setting (n + -0.5) equal to 5.5 and -5.5.Subproblem 1
n + -0.5 = 5.5 Simplifying n + -0.5 = 5.5 Reorder the terms: -0.5 + n = 5.5 Solving -0.5 + n = 5.5 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + n = 5.5 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + n = 5.5 + 0.5 n = 5.5 + 0.5 Combine like terms: 5.5 + 0.5 = 6 n = 6 Simplifying n = 6Subproblem 2
n + -0.5 = -5.5 Simplifying n + -0.5 = -5.5 Reorder the terms: -0.5 + n = -5.5 Solving -0.5 + n = -5.5 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + n = -5.5 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + n = -5.5 + 0.5 n = -5.5 + 0.5 Combine like terms: -5.5 + 0.5 = -5 n = -5 Simplifying n = -5Solution
The solution to the problem is based on the solutions from the subproblems. n = {6, -5}
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