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Simplifying 3n2 + -103n + 96 = 0 Reorder the terms: 96 + -103n + 3n2 = 0 Solving 96 + -103n + 3n2 = 0 Solving for variable 'n'. Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. 32 + -34.33333333n + n2 = 0 Move the constant term to the right: Add '-32' to each side of the equation. 32 + -34.33333333n + -32 + n2 = 0 + -32 Reorder the terms: 32 + -32 + -34.33333333n + n2 = 0 + -32 Combine like terms: 32 + -32 = 0 0 + -34.33333333n + n2 = 0 + -32 -34.33333333n + n2 = 0 + -32 Combine like terms: 0 + -32 = -32 -34.33333333n + n2 = -32 The n term is -34.33333333n. Take half its coefficient (-17.16666667). Square it (294.6944446) and add it to both sides. Add '294.6944446' to each side of the equation. -34.33333333n + 294.6944446 + n2 = -32 + 294.6944446 Reorder the terms: 294.6944446 + -34.33333333n + n2 = -32 + 294.6944446 Combine like terms: -32 + 294.6944446 = 262.6944446 294.6944446 + -34.33333333n + n2 = 262.6944446 Factor a perfect square on the left side: (n + -17.16666667)(n + -17.16666667) = 262.6944446 Calculate the square root of the right side: 16.207851326 Break this problem into two subproblems by setting (n + -17.16666667) equal to 16.207851326 and -16.207851326.Subproblem 1
n + -17.16666667 = 16.207851326 Simplifying n + -17.16666667 = 16.207851326 Reorder the terms: -17.16666667 + n = 16.207851326 Solving -17.16666667 + n = 16.207851326 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '17.16666667' to each side of the equation. -17.16666667 + 17.16666667 + n = 16.207851326 + 17.16666667 Combine like terms: -17.16666667 + 17.16666667 = 0.00000000 0.00000000 + n = 16.207851326 + 17.16666667 n = 16.207851326 + 17.16666667 Combine like terms: 16.207851326 + 17.16666667 = 33.374517996 n = 33.374517996 Simplifying n = 33.374517996Subproblem 2
n + -17.16666667 = -16.207851326 Simplifying n + -17.16666667 = -16.207851326 Reorder the terms: -17.16666667 + n = -16.207851326 Solving -17.16666667 + n = -16.207851326 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '17.16666667' to each side of the equation. -17.16666667 + 17.16666667 + n = -16.207851326 + 17.16666667 Combine like terms: -17.16666667 + 17.16666667 = 0.00000000 0.00000000 + n = -16.207851326 + 17.16666667 n = -16.207851326 + 17.16666667 Combine like terms: -16.207851326 + 17.16666667 = 0.958815344 n = 0.958815344 Simplifying n = 0.958815344Solution
The solution to the problem is based on the solutions from the subproblems. n = {33.374517996, 0.958815344}
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