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Simplifying 750 + 22n(n + -12) = 650 + 30(n + -10) Reorder the terms: 750 + 22n(-12 + n) = 650 + 30(n + -10) 750 + (-12 * 22n + n * 22n) = 650 + 30(n + -10) 750 + (-264n + 22n2) = 650 + 30(n + -10) Reorder the terms: 750 + -264n + 22n2 = 650 + 30(-10 + n) 750 + -264n + 22n2 = 650 + (-10 * 30 + n * 30) 750 + -264n + 22n2 = 650 + (-300 + 30n) Combine like terms: 650 + -300 = 350 750 + -264n + 22n2 = 350 + 30n Solving 750 + -264n + 22n2 = 350 + 30n Solving for variable 'n'. Reorder the terms: 750 + -350 + -264n + -30n + 22n2 = 350 + 30n + -350 + -30n Combine like terms: 750 + -350 = 400 400 + -264n + -30n + 22n2 = 350 + 30n + -350 + -30n Combine like terms: -264n + -30n = -294n 400 + -294n + 22n2 = 350 + 30n + -350 + -30n Reorder the terms: 400 + -294n + 22n2 = 350 + -350 + 30n + -30n Combine like terms: 350 + -350 = 0 400 + -294n + 22n2 = 0 + 30n + -30n 400 + -294n + 22n2 = 30n + -30n Combine like terms: 30n + -30n = 0 400 + -294n + 22n2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(200 + -147n + 11n2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(200 + -147n + 11n2)' equal to zero and attempt to solve: Simplifying 200 + -147n + 11n2 = 0 Solving 200 + -147n + 11n2 = 0 Begin completing the square. Divide all terms by 11 the coefficient of the squared term: Divide each side by '11'. 18.18181818 + -13.36363636n + n2 = 0 Move the constant term to the right: Add '-18.18181818' to each side of the equation. 18.18181818 + -13.36363636n + -18.18181818 + n2 = 0 + -18.18181818 Reorder the terms: 18.18181818 + -18.18181818 + -13.36363636n + n2 = 0 + -18.18181818 Combine like terms: 18.18181818 + -18.18181818 = 0.00000000 0.00000000 + -13.36363636n + n2 = 0 + -18.18181818 -13.36363636n + n2 = 0 + -18.18181818 Combine like terms: 0 + -18.18181818 = -18.18181818 -13.36363636n + n2 = -18.18181818 The n term is -13.36363636n. Take half its coefficient (-6.68181818). Square it (44.64669419) and add it to both sides. Add '44.64669419' to each side of the equation. -13.36363636n + 44.64669419 + n2 = -18.18181818 + 44.64669419 Reorder the terms: 44.64669419 + -13.36363636n + n2 = -18.18181818 + 44.64669419 Combine like terms: -18.18181818 + 44.64669419 = 26.46487601 44.64669419 + -13.36363636n + n2 = 26.46487601 Factor a perfect square on the left side: (n + -6.68181818)(n + -6.68181818) = 26.46487601 Calculate the square root of the right side: 5.144402396 Break this problem into two subproblems by setting (n + -6.68181818) equal to 5.144402396 and -5.144402396.Subproblem 1
n + -6.68181818 = 5.144402396 Simplifying n + -6.68181818 = 5.144402396 Reorder the terms: -6.68181818 + n = 5.144402396 Solving -6.68181818 + n = 5.144402396 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '6.68181818' to each side of the equation. -6.68181818 + 6.68181818 + n = 5.144402396 + 6.68181818 Combine like terms: -6.68181818 + 6.68181818 = 0.00000000 0.00000000 + n = 5.144402396 + 6.68181818 n = 5.144402396 + 6.68181818 Combine like terms: 5.144402396 + 6.68181818 = 11.826220576 n = 11.826220576 Simplifying n = 11.826220576Subproblem 2
n + -6.68181818 = -5.144402396 Simplifying n + -6.68181818 = -5.144402396 Reorder the terms: -6.68181818 + n = -5.144402396 Solving -6.68181818 + n = -5.144402396 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '6.68181818' to each side of the equation. -6.68181818 + 6.68181818 + n = -5.144402396 + 6.68181818 Combine like terms: -6.68181818 + 6.68181818 = 0.00000000 0.00000000 + n = -5.144402396 + 6.68181818 n = -5.144402396 + 6.68181818 Combine like terms: -5.144402396 + 6.68181818 = 1.537415784 n = 1.537415784 Simplifying n = 1.537415784Solution
The solution to the problem is based on the solutions from the subproblems. n = {11.826220576, 1.537415784}Solution
n = {11.826220576, 1.537415784}
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