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Simplifying 4n2 + n2 = 12n + 3 Combine like terms: 4n2 + n2 = 5n2 5n2 = 12n + 3 Reorder the terms: 5n2 = 3 + 12n Solving 5n2 = 3 + 12n Solving for variable 'n'. Reorder the terms: -3 + -12n + 5n2 = 3 + 12n + -3 + -12n Reorder the terms: -3 + -12n + 5n2 = 3 + -3 + 12n + -12n Combine like terms: 3 + -3 = 0 -3 + -12n + 5n2 = 0 + 12n + -12n -3 + -12n + 5n2 = 12n + -12n Combine like terms: 12n + -12n = 0 -3 + -12n + 5n2 = 0 Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. -0.6 + -2.4n + n2 = 0 Move the constant term to the right: Add '0.6' to each side of the equation. -0.6 + -2.4n + 0.6 + n2 = 0 + 0.6 Reorder the terms: -0.6 + 0.6 + -2.4n + n2 = 0 + 0.6 Combine like terms: -0.6 + 0.6 = 0.0 0.0 + -2.4n + n2 = 0 + 0.6 -2.4n + n2 = 0 + 0.6 Combine like terms: 0 + 0.6 = 0.6 -2.4n + n2 = 0.6 The n term is -2.4n. Take half its coefficient (-1.2). Square it (1.44) and add it to both sides. Add '1.44' to each side of the equation. -2.4n + 1.44 + n2 = 0.6 + 1.44 Reorder the terms: 1.44 + -2.4n + n2 = 0.6 + 1.44 Combine like terms: 0.6 + 1.44 = 2.04 1.44 + -2.4n + n2 = 2.04 Factor a perfect square on the left side: (n + -1.2)(n + -1.2) = 2.04 Calculate the square root of the right side: 1.428285686 Break this problem into two subproblems by setting (n + -1.2) equal to 1.428285686 and -1.428285686.Subproblem 1
n + -1.2 = 1.428285686 Simplifying n + -1.2 = 1.428285686 Reorder the terms: -1.2 + n = 1.428285686 Solving -1.2 + n = 1.428285686 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.2' to each side of the equation. -1.2 + 1.2 + n = 1.428285686 + 1.2 Combine like terms: -1.2 + 1.2 = 0.0 0.0 + n = 1.428285686 + 1.2 n = 1.428285686 + 1.2 Combine like terms: 1.428285686 + 1.2 = 2.628285686 n = 2.628285686 Simplifying n = 2.628285686Subproblem 2
n + -1.2 = -1.428285686 Simplifying n + -1.2 = -1.428285686 Reorder the terms: -1.2 + n = -1.428285686 Solving -1.2 + n = -1.428285686 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.2' to each side of the equation. -1.2 + 1.2 + n = -1.428285686 + 1.2 Combine like terms: -1.2 + 1.2 = 0.0 0.0 + n = -1.428285686 + 1.2 n = -1.428285686 + 1.2 Combine like terms: -1.428285686 + 1.2 = -0.228285686 n = -0.228285686 Simplifying n = -0.228285686Solution
The solution to the problem is based on the solutions from the subproblems. n = {2.628285686, -0.228285686}
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