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Simplifying (6n2 + -4n + 8) + -1(10n2 + 8n + -4) = 0 Reorder the terms: (8 + -4n + 6n2) + -1(10n2 + 8n + -4) = 0 Remove parenthesis around (8 + -4n + 6n2) 8 + -4n + 6n2 + -1(10n2 + 8n + -4) = 0 Reorder the terms: 8 + -4n + 6n2 + -1(-4 + 8n + 10n2) = 0 8 + -4n + 6n2 + (-4 * -1 + 8n * -1 + 10n2 * -1) = 0 8 + -4n + 6n2 + (4 + -8n + -10n2) = 0 Reorder the terms: 8 + 4 + -4n + -8n + 6n2 + -10n2 = 0 Combine like terms: 8 + 4 = 12 12 + -4n + -8n + 6n2 + -10n2 = 0 Combine like terms: -4n + -8n = -12n 12 + -12n + 6n2 + -10n2 = 0 Combine like terms: 6n2 + -10n2 = -4n2 12 + -12n + -4n2 = 0 Solving 12 + -12n + -4n2 = 0 Solving for variable 'n'. Factor out the Greatest Common Factor (GCF), '4'. 4(3 + -3n + -1n2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(3 + -3n + -1n2)' equal to zero and attempt to solve: Simplifying 3 + -3n + -1n2 = 0 Solving 3 + -3n + -1n2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -3 + 3n + n2 = 0 Move the constant term to the right: Add '3' to each side of the equation. -3 + 3n + 3 + n2 = 0 + 3 Reorder the terms: -3 + 3 + 3n + n2 = 0 + 3 Combine like terms: -3 + 3 = 0 0 + 3n + n2 = 0 + 3 3n + n2 = 0 + 3 Combine like terms: 0 + 3 = 3 3n + n2 = 3 The n term is 3n. Take half its coefficient (1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. 3n + 2.25 + n2 = 3 + 2.25 Reorder the terms: 2.25 + 3n + n2 = 3 + 2.25 Combine like terms: 3 + 2.25 = 5.25 2.25 + 3n + n2 = 5.25 Factor a perfect square on the left side: (n + 1.5)(n + 1.5) = 5.25 Calculate the square root of the right side: 2.291287847 Break this problem into two subproblems by setting (n + 1.5) equal to 2.291287847 and -2.291287847.Subproblem 1
n + 1.5 = 2.291287847 Simplifying n + 1.5 = 2.291287847 Reorder the terms: 1.5 + n = 2.291287847 Solving 1.5 + n = 2.291287847 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + n = 2.291287847 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + n = 2.291287847 + -1.5 n = 2.291287847 + -1.5 Combine like terms: 2.291287847 + -1.5 = 0.791287847 n = 0.791287847 Simplifying n = 0.791287847Subproblem 2
n + 1.5 = -2.291287847 Simplifying n + 1.5 = -2.291287847 Reorder the terms: 1.5 + n = -2.291287847 Solving 1.5 + n = -2.291287847 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + n = -2.291287847 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + n = -2.291287847 + -1.5 n = -2.291287847 + -1.5 Combine like terms: -2.291287847 + -1.5 = -3.791287847 n = -3.791287847 Simplifying n = -3.791287847Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.791287847, -3.791287847}Solution
n = {0.791287847, -3.791287847}
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