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Simplifying 6n + 19 = 4 + n2 Reorder the terms: 19 + 6n = 4 + n2 Solving 19 + 6n = 4 + n2 Solving for variable 'n'. Reorder the terms: 19 + -4 + 6n + -1n2 = 4 + n2 + -4 + -1n2 Combine like terms: 19 + -4 = 15 15 + 6n + -1n2 = 4 + n2 + -4 + -1n2 Reorder the terms: 15 + 6n + -1n2 = 4 + -4 + n2 + -1n2 Combine like terms: 4 + -4 = 0 15 + 6n + -1n2 = 0 + n2 + -1n2 15 + 6n + -1n2 = n2 + -1n2 Combine like terms: n2 + -1n2 = 0 15 + 6n + -1n2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -15 + -6n + n2 = 0 Move the constant term to the right: Add '15' to each side of the equation. -15 + -6n + 15 + n2 = 0 + 15 Reorder the terms: -15 + 15 + -6n + n2 = 0 + 15 Combine like terms: -15 + 15 = 0 0 + -6n + n2 = 0 + 15 -6n + n2 = 0 + 15 Combine like terms: 0 + 15 = 15 -6n + n2 = 15 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 = 15 + 9 Reorder the terms: 9 + -6n + n2 = 15 + 9 Combine like terms: 15 + 9 = 24 9 + -6n + n2 = 24 Factor a perfect square on the left side: (n + -3)(n + -3) = 24 Calculate the square root of the right side: 4.898979486 Break this problem into two subproblems by setting (n + -3) equal to 4.898979486 and -4.898979486.Subproblem 1
n + -3 = 4.898979486 Simplifying n + -3 = 4.898979486 Reorder the terms: -3 + n = 4.898979486 Solving -3 + n = 4.898979486 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 = 4.898979486 + 3 Combine like terms: -3 + 3 = 0 0 + n = 4.898979486 + 3 n = 4.898979486 + 3 Combine like terms: 4.898979486 + 3 = 7.898979486 n = 7.898979486 Simplifying n = 7.898979486Subproblem 2
n + -3 = -4.898979486 Simplifying n + -3 = -4.898979486 Reorder the terms: -3 + n = -4.898979486 Solving -3 + n = -4.898979486 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 = -4.898979486 + 3 Combine like terms: -3 + 3 = 0 0 + n = -4.898979486 + 3 n = -4.898979486 + 3 Combine like terms: -4.898979486 + 3 = -1.898979486 n = -1.898979486 Simplifying n = -1.898979486Solution
The solution to the problem is based on the solutions from the subproblems. n = {7.898979486, -1.898979486}
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