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Simplifying 3n(7n + -20) = 90 Reorder the terms: 3n(-20 + 7n) = 90 (-20 * 3n + 7n * 3n) = 90 (-60n + 21n2) = 90 Solving -60n + 21n2 = 90 Solving for variable 'n'. Reorder the terms: -90 + -60n + 21n2 = 90 + -90 Combine like terms: 90 + -90 = 0 -90 + -60n + 21n2 = 0 Factor out the Greatest Common Factor (GCF), '3'. 3(-30 + -20n + 7n2) = 0 Ignore the factor 3.Subproblem 1
Set the factor '(-30 + -20n + 7n2)' equal to zero and attempt to solve: Simplifying -30 + -20n + 7n2 = 0 Solving -30 + -20n + 7n2 = 0 Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. -4.285714286 + -2.857142857n + n2 = 0 Move the constant term to the right: Add '4.285714286' to each side of the equation. -4.285714286 + -2.857142857n + 4.285714286 + n2 = 0 + 4.285714286 Reorder the terms: -4.285714286 + 4.285714286 + -2.857142857n + n2 = 0 + 4.285714286 Combine like terms: -4.285714286 + 4.285714286 = 0.000000000 0.000000000 + -2.857142857n + n2 = 0 + 4.285714286 -2.857142857n + n2 = 0 + 4.285714286 Combine like terms: 0 + 4.285714286 = 4.285714286 -2.857142857n + n2 = 4.285714286 The n term is -2.857142857n. Take half its coefficient (-1.428571429). Square it (2.040816328) and add it to both sides. Add '2.040816328' to each side of the equation. -2.857142857n + 2.040816328 + n2 = 4.285714286 + 2.040816328 Reorder the terms: 2.040816328 + -2.857142857n + n2 = 4.285714286 + 2.040816328 Combine like terms: 4.285714286 + 2.040816328 = 6.326530614 2.040816328 + -2.857142857n + n2 = 6.326530614 Factor a perfect square on the left side: (n + -1.428571429)(n + -1.428571429) = 6.326530614 Calculate the square root of the right side: 2.515259552 Break this problem into two subproblems by setting (n + -1.428571429) equal to 2.515259552 and -2.515259552.Subproblem 1
n + -1.428571429 = 2.515259552 Simplifying n + -1.428571429 = 2.515259552 Reorder the terms: -1.428571429 + n = 2.515259552 Solving -1.428571429 + n = 2.515259552 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.428571429' to each side of the equation. -1.428571429 + 1.428571429 + n = 2.515259552 + 1.428571429 Combine like terms: -1.428571429 + 1.428571429 = 0.000000000 0.000000000 + n = 2.515259552 + 1.428571429 n = 2.515259552 + 1.428571429 Combine like terms: 2.515259552 + 1.428571429 = 3.943830981 n = 3.943830981 Simplifying n = 3.943830981Subproblem 2
n + -1.428571429 = -2.515259552 Simplifying n + -1.428571429 = -2.515259552 Reorder the terms: -1.428571429 + n = -2.515259552 Solving -1.428571429 + n = -2.515259552 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.428571429' to each side of the equation. -1.428571429 + 1.428571429 + n = -2.515259552 + 1.428571429 Combine like terms: -1.428571429 + 1.428571429 = 0.000000000 0.000000000 + n = -2.515259552 + 1.428571429 n = -2.515259552 + 1.428571429 Combine like terms: -2.515259552 + 1.428571429 = -1.086688123 n = -1.086688123 Simplifying n = -1.086688123Solution
The solution to the problem is based on the solutions from the subproblems. n = {3.943830981, -1.086688123}Solution
n = {3.943830981, -1.086688123}
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