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Simplifying (n + -4) * 4.3n + -2.1 = 15 Reorder the terms: (-4 + n) * 4.3n + -2.1 = 15 Reorder the terms for easier multiplication: 4.3n(-4 + n) + -2.1 = 15 (-4 * 4.3n + n * 4.3n) + -2.1 = 15 (-17.2n + 4.3n2) + -2.1 = 15 Reorder the terms: -2.1 + -17.2n + 4.3n2 = 15 Solving -2.1 + -17.2n + 4.3n2 = 15 Solving for variable 'n'. Reorder the terms: -2.1 + -15 + -17.2n + 4.3n2 = 15 + -15 Combine like terms: -2.1 + -15 = -17.1 -17.1 + -17.2n + 4.3n2 = 15 + -15 Combine like terms: 15 + -15 = 0 -17.1 + -17.2n + 4.3n2 = 0 Begin completing the square. Divide all terms by 4.3 the coefficient of the squared term: Divide each side by '4.3'. -3.976744186 + -4n + n2 = 0 Move the constant term to the right: Add '3.976744186' to each side of the equation. -3.976744186 + -4n + 3.976744186 + n2 = 0 + 3.976744186 Reorder the terms: -3.976744186 + 3.976744186 + -4n + n2 = 0 + 3.976744186 Combine like terms: -3.976744186 + 3.976744186 = 0.000000000 0.000000000 + -4n + n2 = 0 + 3.976744186 -4n + n2 = 0 + 3.976744186 Combine like terms: 0 + 3.976744186 = 3.976744186 -4n + n2 = 3.976744186 The n term is -4n. Take half its coefficient (-2). Square it (4) and add it to both sides. Add '4' to each side of the equation. -4n + 4 + n2 = 3.976744186 + 4 Reorder the terms: 4 + -4n + n2 = 3.976744186 + 4 Combine like terms: 3.976744186 + 4 = 7.976744186 4 + -4n + n2 = 7.976744186 Factor a perfect square on the left side: (n + -2)(n + -2) = 7.976744186 Calculate the square root of the right side: 2.824313047 Break this problem into two subproblems by setting (n + -2) equal to 2.824313047 and -2.824313047.Subproblem 1
n + -2 = 2.824313047 Simplifying n + -2 = 2.824313047 Reorder the terms: -2 + n = 2.824313047 Solving -2 + n = 2.824313047 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + n = 2.824313047 + 2 Combine like terms: -2 + 2 = 0 0 + n = 2.824313047 + 2 n = 2.824313047 + 2 Combine like terms: 2.824313047 + 2 = 4.824313047 n = 4.824313047 Simplifying n = 4.824313047Subproblem 2
n + -2 = -2.824313047 Simplifying n + -2 = -2.824313047 Reorder the terms: -2 + n = -2.824313047 Solving -2 + n = -2.824313047 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + n = -2.824313047 + 2 Combine like terms: -2 + 2 = 0 0 + n = -2.824313047 + 2 n = -2.824313047 + 2 Combine like terms: -2.824313047 + 2 = -0.824313047 n = -0.824313047 Simplifying n = -0.824313047Solution
The solution to the problem is based on the solutions from the subproblems. n = {4.824313047, -0.824313047}
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