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Simplifying 8n2 + 9n + -25 = -9 Reorder the terms: -25 + 9n + 8n2 = -9 Solving -25 + 9n + 8n2 = -9 Solving for variable 'n'. Reorder the terms: -25 + 9 + 9n + 8n2 = -9 + 9 Combine like terms: -25 + 9 = -16 -16 + 9n + 8n2 = -9 + 9 Combine like terms: -9 + 9 = 0 -16 + 9n + 8n2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. -2 + 1.125n + n2 = 0 Move the constant term to the right: Add '2' to each side of the equation. -2 + 1.125n + 2 + n2 = 0 + 2 Reorder the terms: -2 + 2 + 1.125n + n2 = 0 + 2 Combine like terms: -2 + 2 = 0 0 + 1.125n + n2 = 0 + 2 1.125n + n2 = 0 + 2 Combine like terms: 0 + 2 = 2 1.125n + n2 = 2 The n term is 1.125n. Take half its coefficient (0.5625). Square it (0.31640625) and add it to both sides. Add '0.31640625' to each side of the equation. 1.125n + 0.31640625 + n2 = 2 + 0.31640625 Reorder the terms: 0.31640625 + 1.125n + n2 = 2 + 0.31640625 Combine like terms: 2 + 0.31640625 = 2.31640625 0.31640625 + 1.125n + n2 = 2.31640625 Factor a perfect square on the left side: (n + 0.5625)(n + 0.5625) = 2.31640625 Calculate the square root of the right side: 1.521974458 Break this problem into two subproblems by setting (n + 0.5625) equal to 1.521974458 and -1.521974458.Subproblem 1
n + 0.5625 = 1.521974458 Simplifying n + 0.5625 = 1.521974458 Reorder the terms: 0.5625 + n = 1.521974458 Solving 0.5625 + n = 1.521974458 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.5625' to each side of the equation. 0.5625 + -0.5625 + n = 1.521974458 + -0.5625 Combine like terms: 0.5625 + -0.5625 = 0.0000 0.0000 + n = 1.521974458 + -0.5625 n = 1.521974458 + -0.5625 Combine like terms: 1.521974458 + -0.5625 = 0.959474458 n = 0.959474458 Simplifying n = 0.959474458Subproblem 2
n + 0.5625 = -1.521974458 Simplifying n + 0.5625 = -1.521974458 Reorder the terms: 0.5625 + n = -1.521974458 Solving 0.5625 + n = -1.521974458 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.5625' to each side of the equation. 0.5625 + -0.5625 + n = -1.521974458 + -0.5625 Combine like terms: 0.5625 + -0.5625 = 0.0000 0.0000 + n = -1.521974458 + -0.5625 n = -1.521974458 + -0.5625 Combine like terms: -1.521974458 + -0.5625 = -2.084474458 n = -2.084474458 Simplifying n = -2.084474458Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.959474458, -2.084474458}
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