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Simplifying 4n2 = 7n + -1 Reorder the terms: 4n2 = -1 + 7n Solving 4n2 = -1 + 7n Solving for variable 'n'. Reorder the terms: 1 + -7n + 4n2 = -1 + 7n + 1 + -7n Reorder the terms: 1 + -7n + 4n2 = -1 + 1 + 7n + -7n Combine like terms: -1 + 1 = 0 1 + -7n + 4n2 = 0 + 7n + -7n 1 + -7n + 4n2 = 7n + -7n Combine like terms: 7n + -7n = 0 1 + -7n + 4n2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. 0.25 + -1.75n + n2 = 0 Move the constant term to the right: Add '-0.25' to each side of the equation. 0.25 + -1.75n + -0.25 + n2 = 0 + -0.25 Reorder the terms: 0.25 + -0.25 + -1.75n + n2 = 0 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + -1.75n + n2 = 0 + -0.25 -1.75n + n2 = 0 + -0.25 Combine like terms: 0 + -0.25 = -0.25 -1.75n + n2 = -0.25 The n term is -1.75n. Take half its coefficient (-0.875). Square it (0.765625) and add it to both sides. Add '0.765625' to each side of the equation. -1.75n + 0.765625 + n2 = -0.25 + 0.765625 Reorder the terms: 0.765625 + -1.75n + n2 = -0.25 + 0.765625 Combine like terms: -0.25 + 0.765625 = 0.515625 0.765625 + -1.75n + n2 = 0.515625 Factor a perfect square on the left side: (n + -0.875)(n + -0.875) = 0.515625 Calculate the square root of the right side: 0.718070331 Break this problem into two subproblems by setting (n + -0.875) equal to 0.718070331 and -0.718070331.Subproblem 1
n + -0.875 = 0.718070331 Simplifying n + -0.875 = 0.718070331 Reorder the terms: -0.875 + n = 0.718070331 Solving -0.875 + n = 0.718070331 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.875' to each side of the equation. -0.875 + 0.875 + n = 0.718070331 + 0.875 Combine like terms: -0.875 + 0.875 = 0.000 0.000 + n = 0.718070331 + 0.875 n = 0.718070331 + 0.875 Combine like terms: 0.718070331 + 0.875 = 1.593070331 n = 1.593070331 Simplifying n = 1.593070331Subproblem 2
n + -0.875 = -0.718070331 Simplifying n + -0.875 = -0.718070331 Reorder the terms: -0.875 + n = -0.718070331 Solving -0.875 + n = -0.718070331 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.875' to each side of the equation. -0.875 + 0.875 + n = -0.718070331 + 0.875 Combine like terms: -0.875 + 0.875 = 0.000 0.000 + n = -0.718070331 + 0.875 n = -0.718070331 + 0.875 Combine like terms: -0.718070331 + 0.875 = 0.156929669 n = 0.156929669 Simplifying n = 0.156929669Solution
The solution to the problem is based on the solutions from the subproblems. n = {1.593070331, 0.156929669}
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