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Simplifying 5n2 + -3n + -10 = 15 Reorder the terms: -10 + -3n + 5n2 = 15 Solving -10 + -3n + 5n2 = 15 Solving for variable 'n'. Reorder the terms: -10 + -15 + -3n + 5n2 = 15 + -15 Combine like terms: -10 + -15 = -25 -25 + -3n + 5n2 = 15 + -15 Combine like terms: 15 + -15 = 0 -25 + -3n + 5n2 = 0 Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. -5 + -0.6n + n2 = 0 Move the constant term to the right: Add '5' to each side of the equation. -5 + -0.6n + 5 + n2 = 0 + 5 Reorder the terms: -5 + 5 + -0.6n + n2 = 0 + 5 Combine like terms: -5 + 5 = 0 0 + -0.6n + n2 = 0 + 5 -0.6n + n2 = 0 + 5 Combine like terms: 0 + 5 = 5 -0.6n + n2 = 5 The n term is -0.6n. Take half its coefficient (-0.3). Square it (0.09) and add it to both sides. Add '0.09' to each side of the equation. -0.6n + 0.09 + n2 = 5 + 0.09 Reorder the terms: 0.09 + -0.6n + n2 = 5 + 0.09 Combine like terms: 5 + 0.09 = 5.09 0.09 + -0.6n + n2 = 5.09 Factor a perfect square on the left side: (n + -0.3)(n + -0.3) = 5.09 Calculate the square root of the right side: 2.256102835 Break this problem into two subproblems by setting (n + -0.3) equal to 2.256102835 and -2.256102835.Subproblem 1
n + -0.3 = 2.256102835 Simplifying n + -0.3 = 2.256102835 Reorder the terms: -0.3 + n = 2.256102835 Solving -0.3 + n = 2.256102835 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.3' to each side of the equation. -0.3 + 0.3 + n = 2.256102835 + 0.3 Combine like terms: -0.3 + 0.3 = 0.0 0.0 + n = 2.256102835 + 0.3 n = 2.256102835 + 0.3 Combine like terms: 2.256102835 + 0.3 = 2.556102835 n = 2.556102835 Simplifying n = 2.556102835Subproblem 2
n + -0.3 = -2.256102835 Simplifying n + -0.3 = -2.256102835 Reorder the terms: -0.3 + n = -2.256102835 Solving -0.3 + n = -2.256102835 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.3' to each side of the equation. -0.3 + 0.3 + n = -2.256102835 + 0.3 Combine like terms: -0.3 + 0.3 = 0.0 0.0 + n = -2.256102835 + 0.3 n = -2.256102835 + 0.3 Combine like terms: -2.256102835 + 0.3 = -1.956102835 n = -1.956102835 Simplifying n = -1.956102835Solution
The solution to the problem is based on the solutions from the subproblems. n = {2.556102835, -1.956102835}
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