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Simplifying 4n2 + 8n + 3 = 10 Reorder the terms: 3 + 8n + 4n2 = 10 Solving 3 + 8n + 4n2 = 10 Solving for variable 'n'. Reorder the terms: 3 + -10 + 8n + 4n2 = 10 + -10 Combine like terms: 3 + -10 = -7 -7 + 8n + 4n2 = 10 + -10 Combine like terms: 10 + -10 = 0 -7 + 8n + 4n2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -1.75 + 2n + n2 = 0 Move the constant term to the right: Add '1.75' to each side of the equation. -1.75 + 2n + 1.75 + n2 = 0 + 1.75 Reorder the terms: -1.75 + 1.75 + 2n + n2 = 0 + 1.75 Combine like terms: -1.75 + 1.75 = 0.00 0.00 + 2n + n2 = 0 + 1.75 2n + n2 = 0 + 1.75 Combine like terms: 0 + 1.75 = 1.75 2n + n2 = 1.75 The n term is 2n. Take half its coefficient (1). Square it (1) and add it to both sides. Add '1' to each side of the equation. 2n + 1 + n2 = 1.75 + 1 Reorder the terms: 1 + 2n + n2 = 1.75 + 1 Combine like terms: 1.75 + 1 = 2.75 1 + 2n + n2 = 2.75 Factor a perfect square on the left side: (n + 1)(n + 1) = 2.75 Calculate the square root of the right side: 1.658312395 Break this problem into two subproblems by setting (n + 1) equal to 1.658312395 and -1.658312395.Subproblem 1
n + 1 = 1.658312395 Simplifying n + 1 = 1.658312395 Reorder the terms: 1 + n = 1.658312395 Solving 1 + n = 1.658312395 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + n = 1.658312395 + -1 Combine like terms: 1 + -1 = 0 0 + n = 1.658312395 + -1 n = 1.658312395 + -1 Combine like terms: 1.658312395 + -1 = 0.658312395 n = 0.658312395 Simplifying n = 0.658312395Subproblem 2
n + 1 = -1.658312395 Simplifying n + 1 = -1.658312395 Reorder the terms: 1 + n = -1.658312395 Solving 1 + n = -1.658312395 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + n = -1.658312395 + -1 Combine like terms: 1 + -1 = 0 0 + n = -1.658312395 + -1 n = -1.658312395 + -1 Combine like terms: -1.658312395 + -1 = -2.658312395 n = -2.658312395 Simplifying n = -2.658312395Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.658312395, -2.658312395}
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