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Simplifying 2n2 = 43 + 12n Solving 2n2 = 43 + 12n Solving for variable 'n'. Reorder the terms: -43 + -12n + 2n2 = 43 + 12n + -43 + -12n Reorder the terms: -43 + -12n + 2n2 = 43 + -43 + 12n + -12n Combine like terms: 43 + -43 = 0 -43 + -12n + 2n2 = 0 + 12n + -12n -43 + -12n + 2n2 = 12n + -12n Combine like terms: 12n + -12n = 0 -43 + -12n + 2n2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -21.5 + -6n + n2 = 0 Move the constant term to the right: Add '21.5' to each side of the equation. -21.5 + -6n + 21.5 + n2 = 0 + 21.5 Reorder the terms: -21.5 + 21.5 + -6n + n2 = 0 + 21.5 Combine like terms: -21.5 + 21.5 = 0.0 0.0 + -6n + n2 = 0 + 21.5 -6n + n2 = 0 + 21.5 Combine like terms: 0 + 21.5 = 21.5 -6n + n2 = 21.5 The n term is -6n. Take half its coefficient (-3). Square it (9) and add it to both sides. Add '9' to each side of the equation. -6n + 9 + n2 = 21.5 + 9 Reorder the terms: 9 + -6n + n2 = 21.5 + 9 Combine like terms: 21.5 + 9 = 30.5 9 + -6n + n2 = 30.5 Factor a perfect square on the left side: (n + -3)(n + -3) = 30.5 Calculate the square root of the right side: 5.522680509 Break this problem into two subproblems by setting (n + -3) equal to 5.522680509 and -5.522680509.Subproblem 1
n + -3 = 5.522680509 Simplifying n + -3 = 5.522680509 Reorder the terms: -3 + n = 5.522680509 Solving -3 + n = 5.522680509 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + n = 5.522680509 + 3 Combine like terms: -3 + 3 = 0 0 + n = 5.522680509 + 3 n = 5.522680509 + 3 Combine like terms: 5.522680509 + 3 = 8.522680509 n = 8.522680509 Simplifying n = 8.522680509Subproblem 2
n + -3 = -5.522680509 Simplifying n + -3 = -5.522680509 Reorder the terms: -3 + n = -5.522680509 Solving -3 + n = -5.522680509 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + n = -5.522680509 + 3 Combine like terms: -3 + 3 = 0 0 + n = -5.522680509 + 3 n = -5.522680509 + 3 Combine like terms: -5.522680509 + 3 = -2.522680509 n = -2.522680509 Simplifying n = -2.522680509Solution
The solution to the problem is based on the solutions from the subproblems. n = {8.522680509, -2.522680509}
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