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Simplifying 7n2 + -8 = -12n Reorder the terms: -8 + 7n2 = -12n Solving -8 + 7n2 = -12n Solving for variable 'n'. Reorder the terms: -8 + 12n + 7n2 = -12n + 12n Combine like terms: -12n + 12n = 0 -8 + 12n + 7n2 = 0 Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. -1.142857143 + 1.714285714n + n2 = 0 Move the constant term to the right: Add '1.142857143' to each side of the equation. -1.142857143 + 1.714285714n + 1.142857143 + n2 = 0 + 1.142857143 Reorder the terms: -1.142857143 + 1.142857143 + 1.714285714n + n2 = 0 + 1.142857143 Combine like terms: -1.142857143 + 1.142857143 = 0.000000000 0.000000000 + 1.714285714n + n2 = 0 + 1.142857143 1.714285714n + n2 = 0 + 1.142857143 Combine like terms: 0 + 1.142857143 = 1.142857143 1.714285714n + n2 = 1.142857143 The n term is 1.714285714n. Take half its coefficient (0.857142857). Square it (0.7346938773) and add it to both sides. Add '0.7346938773' to each side of the equation. 1.714285714n + 0.7346938773 + n2 = 1.142857143 + 0.7346938773 Reorder the terms: 0.7346938773 + 1.714285714n + n2 = 1.142857143 + 0.7346938773 Combine like terms: 1.142857143 + 0.7346938773 = 1.8775510203 0.7346938773 + 1.714285714n + n2 = 1.8775510203 Factor a perfect square on the left side: (n + 0.857142857)(n + 0.857142857) = 1.8775510203 Calculate the square root of the right side: 1.370237578 Break this problem into two subproblems by setting (n + 0.857142857) equal to 1.370237578 and -1.370237578.Subproblem 1
n + 0.857142857 = 1.370237578 Simplifying n + 0.857142857 = 1.370237578 Reorder the terms: 0.857142857 + n = 1.370237578 Solving 0.857142857 + n = 1.370237578 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.857142857' to each side of the equation. 0.857142857 + -0.857142857 + n = 1.370237578 + -0.857142857 Combine like terms: 0.857142857 + -0.857142857 = 0.000000000 0.000000000 + n = 1.370237578 + -0.857142857 n = 1.370237578 + -0.857142857 Combine like terms: 1.370237578 + -0.857142857 = 0.513094721 n = 0.513094721 Simplifying n = 0.513094721Subproblem 2
n + 0.857142857 = -1.370237578 Simplifying n + 0.857142857 = -1.370237578 Reorder the terms: 0.857142857 + n = -1.370237578 Solving 0.857142857 + n = -1.370237578 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.857142857' to each side of the equation. 0.857142857 + -0.857142857 + n = -1.370237578 + -0.857142857 Combine like terms: 0.857142857 + -0.857142857 = 0.000000000 0.000000000 + n = -1.370237578 + -0.857142857 n = -1.370237578 + -0.857142857 Combine like terms: -1.370237578 + -0.857142857 = -2.227380435 n = -2.227380435 Simplifying n = -2.227380435Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.513094721, -2.227380435}
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